173.55 6518 rl H hh ^9 The Theory of GENERAL RELATIVITY and GRAVITATION Based on a course of lectures delivered at the Conference on Recent Advances in Physics held at the University of Toronto, in January, 1921 LUDWIK SILBERSTEIN, Ph.D. New York D. VAN NOSTRAND COMPANY Eight Warren Street 1922 Copyright, Canada, 1922 BY THE University of Toronto Press io 8 no R i Musics S575 PREFACE At the Conference on Recent Advances in Physics held in the Physics Laboratory of the University of Toronto from January 5 to 26, 1921, a course on Einstein's Relativity and Gravitation Theory, consisting of fifteen lectures and two colloquia, was delivered by the author. The first six of 1 1 ■ lectures were devoted to what is known as Special Relativity, and the remaining ones to Einstein's General Relativity and Gravitation Theory and to relativistic Electromagnetism. In view of the time limitations only the essentials of these theories were dealt with, due attention, however, being given to the critically conceptual side of the subject. The Uniwr- sity was kind enough to undertake the publication of that part of the course which dealt with general relativistic ques- tions, on the express understanding that my prospective readers should be assumed to be already familiar with the special theory of relativity. In this connection it was sug- gested by Prof. McLennan that those unacquainted with the older theory should be referred to my book of 1914 (The Theory of Relativity, Macmillan, London) and that it would therefore be desirable to make the present volume, as much as possible, uniform in exposition and style with that work. \Vith such requirements in view this little book was shaped, only a few pages at the beginning having been used in re- calling the essentials of the special relativity theory. The treatment, as compared with the Toronto lectures, has been made somewhat more systematic and the subject matter has, here and there, been considerably extended. In this respect the author has been partly influenced by a larger course on Relativity, Gravitation and Electromagnetism delivered, in the time of writing, during the last Summer Quarter at the University of Chicago. Such is especially the case with Chapter III in which care has been taken to give the readers a systematic exposition of the calculus of generally covariant beings called Tensors. The exposition follows here mainly upon Einstein's own presentation of the subject, with the difference, however, that due emphasis has been laid upon the distinction between metrical and non- metrical properties of tensors. But even in this chapter technicalities have been avoided, stress being laid upon the guiding principles of this new, or rather newly revived, and most powerful mathematical method. It seems hard to say whether Einstein's admirable theory has or has not a long life before it in the domain of Physics proper. But indepen- dently of its fate the time applied for studying the Tensor Calculus and acquiring some skill in handling it will be well spent. The plan of the remainder of the book will be sufficiently clear from the titles of the chapters and sections arrayed in the table of Contents. Such matter as seemed for the present too speculative and controversial has been relegated to the x Appendix where, however, also some points concerning the curvature properties of a manifold have found their place, not only as a preparatory to Einstein's cosmological specu- lations but perhaps as a useful supplement to Chapter III. The book is felt to be far from being complete. But as it is, it is hoped that it will give the reader a good insight into the guiding spirit of Einstein's general relativity and gravita- tion theory and enable him to follow without serious diffi- culties the deeper investigations and the more special and extended developments given in the large and rapidly growing number of papers on the subject. Some of my readers will miss, perhaps, in this volume the enthusiastic tone which usually permeates the books and pamphlets that have been written on the subject (with a notable exception of Einstein's own writings). Yet the author is the last man to be blind to the admirable boldness and the severe architectonic beauty of Einstein's theory. But it has seemed that beauties of such a kind are rather enhanced than obscured by the adoption of a sober tone and an apparently cold form of presentation. My thanks are due to Sir Robert Falconer and to Prof. J. C. McLennan for promoting the cause of this publication, to Prof. R. A. Millikan and Prof. Henry G. Gale of the University of Chicago for reading part of the proofs, and to the University of Toronto Press for the care bestowed on my work. L. S. Rochester, N.Y. November 1921. CONTENTS CHAPTER I Special Relativity recalled. Foundations of General Relativity and Gravitation Theory PAGE 1. Inertia] reference systems 1 2. Special relativity principle - 3. Principle of constant light velocity 4. Lorentz transformation. Galilean line-element. Minimal lines and geodesies representing light propagation and motion ot free particles 1 5. Transition to general relativity and gravitation theory. Infini- tesimal equivalence hypothesis and local coordinates 9 6. Gaussian coordinates, and the general line-element 14 7. Light propagation and free-particle motion expressed by the general null-lines and geodesies 17 CHAPTER II The General Relativity Principle. Minimal Lines and Geodesies. Examples. Newton's Equations of Motion as an Approximation 8. Principles of general relativity; general covariance of laws. . . . 22 9. Local and system-velocity of light 25 10. Developed form of geodesies. Christoffel symbols 26 10a. First example: galilean system 29 10b. Second example: rotating system 30 11. Geodesies, and Newtonian equations of motion as an approxi- mation 35 CHAPTER III Elements of Tensor Algebra and Analysis 12. Introductory. Gaussian coordinates .39 13. Contravariant and covariant tensors of rank one or vectors . . . 40 14. Inner or scalar product of two vectors. Zero rank tensors, invariants i'2 15. Outer product. Tensors of rank two, symmetrical and anti- symmetrical. Mixed tensors 43 16. Tensors of any rank 46 17. Contraction. Intrinsic invariants 46 18. Inner multiplication. Differentiation of tensors 48 19. Tensor properties in a metrical field. Quadratic differential form or line-element 50 20. Fundamental tensor. Metrical properties of tensors. Norm and size. Conjugate tensors 52 21. Supplement. Reduced tensor 55 22. Angle and volume. Sub-domains 56 PAGE 23.^Differentiation based on metrics. Covariant derivative or ex- pansion; contra variant derivative. Rotation of a vector. Antisymmetric expansion of a six-vector. Divergence of a six-and of a four-vector 59 24. The Riemann-Christoffel tensor. Riemannian symbols and curvature. Lipschitz's theorem 62 CHAPTER IV The Gravitational Field-equations, and the Tensor of Matter 25. Contracted curvature tensor. Einstein's field-equations outside $$ of matter. Bianchi's identities 69 26. Laplace's equation, and Newton's law, as a first approximation 72 27. The tensor of matter. Einstein's field-equations within matter. Laplace-Poisson's equation as a first approximation. Mean curvature and density of matter. Example 74 28. Equations of Matter. The principles of momentum and of energy. Remarks on conservation 81 29. Hamiltonian Principle 88 30. Gravitational waves. Einstein's approximate integration of the field-equations 90 CHAPTER V Radially Symmetric Field. Perihelion Motion, Bending of Rays, and Spectrum Shift 31. Radially symmetrical solution of the field-equations 92 32. Perihelion of a planet. Mercury's excess 95 33. Deflection of light rays. Results of the Sobral Eclipse Expedi- tion 100 34. Shift of spectrum lines. The atoms as 'natural clocks' 102 CHAPTER VI 35. Generally covariant form of the equations of the electromag- netic field 106 36. The four-potential Ill 37. Orthogonal curvilinear coordinates 113 38. Propagation of electromagnetic waves in a gravitational field . . 115 39. Ponderomotive force and energy tensor of the electromagnetic field 119 APPENDIX A. Manifolds of Constant Curvature 124 B. Einstein's New Field-Equations and Elliptic Space 129 C. Space-Time according to de Sitter 135 D. Gravitational Fields and Electrons 137 Index 138 CHAPTER I. Special Relativity recalled. Foundations of General Relativity and Gravitation Theory. In accordance with the purpose and the origin of this volume* its readers are assumed to have already made them- selves familiar with the essentials of Einstein's older or special Relativity. It will be enough, therefore, to recall here very concisely what of that theory may be conducive to, and even necessary for, a thorough grasping of the structure and the aims of the more general theory, and of the spirit pervading it. 1. First of all, then, out of all thinkable reference- frame- works, the special relativity is concerned only with a certain privileged class of frameworks or systems of reference, the inertial systems. Of these there are <» 3 . If S, say the 'fixed-stars' system, is one of them, any other rigid system S' of coordinate axes moving relatively to 5 with any uniform and purely translational velocity v, in any direction whatever, is again an inertial system or belongs to the same privileged class. And the systems thus derived from S, or from one another,** exhaust the class. Since the size or absolute value of the relative velocity implies one scalar datum, and its direction two more such data, all independent of one another, there is just a triple infinity of inertial systems.f as already stated. Not that the special relativity theory abstains from considering accelerated, i.e. non-uniform motion of particles within any of these systems; but it does not contemplate any frameworks other than the inertial ones as systems of refer- *Cf. Preface. **If S r moves uniformly with respect to 5, and S" with respect to S', so does 5 with respect to 5. If the reader so desires, he may consider this as a postulate. fThe purely spatial orientation of the axes, implying further free data, is irrelevant in the present connection. 1 2 ' Relativity and Gravitation ence, and cannot, nor does it propose to deal with them. It is unable, for instance, to transform the course of phenomena from the 5 system to the spinning Earth or to an accelerated carriage as reference systems. 2. Keeping this in mind, the first main assumption of the older theory, known as the Special Relativity Principle, can be briefly stated by saying that it requires the laws of physical phenomena to be the same whether they are referred to one or to any other inertial system. In short, the maxim of the 1905 — Relativity was: Equal laws for all inertial systems. The italicized words are, mathematically speaking, at first somewhat vague. In fact, they are intended to stand for 'the same form of mathematical equations expressing the laws.' Now, since this implies the use of some magnitudes, such as the coordinates and the time, or the electric and the magnetic vectors (forces), in each of the said systems, the requirement of mathematical 'sameness' remains cloudy until we are told what dictionary is to be used to translate the language of one into that of any other inertial system, or technically, to transform from the non-dashed to the dashed variables. This vagueness, however, soon disappears, giving place to precision, in the next fundamental step of the theory as will be seen presently. The attentive reader might here object by saying that 'sameness of laws' means absence of difference, absence of observable different behaviour (of moving bodies or of electric waves) in passing from an S to an 5 , and that, therefore, to begin with, no mathematical magnitudes or equations are required. But actually we are, perhaps forever, confined to one (approximately) inertial system, our planet, and are thus unable to observe directly the permanence of behaviour in passing to another system of reference. The only way open to us is to proceed, through more or less long chains of abstract reasoning, from the principle of relativity to some observable prediction, and such processes are scarcely practicable without the use of mathematical symbols and equations. 3. The second assumption, called the Principle of Constant Light- velocity, apart from its own importance, provides for the need just explained, its true office in the structure of the theory being to set an example of a 'physical law' which is postulated to satisfy rigorously the first assumption. It Constani Light Velocity nms thus: Liu' 1 ' is propagated, in vacuo, relatively inertial system, with a velocity c, con d equal for all directions, no matter whether the source emitting it is fixed or moving with reaped to that system. This is Bhortl) referred to as uniform and isotropic light propagation in an) inertial system. The light velocity, in empty space, plays th<- part of a universal constant, — which role, however, it will readily give up in generalized relativity. The reader is well acquainted with the mathematical expression of the consequence of these two assumptions (together with a tacit requirement of formal equivalence of any two inertial systems S, S'), to wit, the invariance of the quadratic form c 2 t 2 -x 2 -y--z 2 , where x, y, z are the cartesian co-ordinates and / the time of the ^-system. That is to say, if x', y', 2', t' be the cartesian co-ordinates and the time used in any other inertial system S', (a) should transform into c i r--x n --y" l -z'-. (a') As a matter of fact, what was originally required was that tin- equation (a)=0 should transform into (a') = 0, and this would be satisfied by putting (a') =\ . (a), where X is inde- pendentof x, y, z, t but might be some function of v, the relative velocity of S, S'. This, however, would amount to a dis- tinction between the two systems, at least a formal one. unless X=l. If, therefore, equal rights are claimed not only physically but also formally, mathematically, for all inertial systems, we have (a) = (a'), that is to say, the quadratic form (a) is raised to the dignity of an invariant. There is, certainly, nothing to object to in such a procedure, especially as it carries simplicity with itself. Yet these remarks did not seem superfluous, especially as there is among the relativists a strong tendency to a certain kind of hypostasy of the said quadatic form* (by declaring it to be more intensified more recently in the case of the more general (differ- ential) quadratic form playing a fundamental r61e in the newer relativity theory, as will be ?een hereafter. 4 Relativity and Gravitation 'objective, real or intrinsic' than space-distance or time) just because it "is" invariant, — and forgetting that we have deliberately made it invariant. 4. Meanwhile, returning to the quadratic expression (a), let us write it down for a pair of events infinitesimally near to one another in space and time. Thus, writing X\, x 2 , x 3 , x* for x, y, z, ct, the statements made above can be expressed by saying that the quadratic differential form ds 2 = dxi 2 — dxi 2 —dx2 2 —dxz z (1) should be invariant with respect to the passage from one inertial system 5 to any other such system S f . The differ- ential foim is here preferable to the original one, as it will be helpful in paving the way for general relativity. As is well known, this requirement of invariance gives the rule of transformation of the variables x t into those x\ of the ^'-system, called the Lorentz transformation. If both the Xi and the x\ axes are taken along the line of motion of S' relatively to S, with the velocity v = /3c, if further the x 2 , x s — axes are taken parallel to those of x' 2 , x'z, and if the convention x'x = x' 4=0 for xi = x 4 = is adopted, the Lorentz transforma- tion assumes the familiar form x'i = y(x! — j8x 4 ), x' 2 = x 2 , x'z = x 3 , x' i = y(x i — px 1 ) (2) where y — (1 — /3 2 ) ~~**. Vice versa, we have, by solving (2), Xi = y(x'i+t3x'^, x 2 =x' 2 , Xz = x' z , Xi = y{x' i+fix'i), showing the complete (including the formal) equivalence of the two systems. Let us keep well in mind, for what is to follow, that this transformation is a linear one, with constant co- efficients, and that special relativity, concerned with inertial systems only, does not contemplate any other space-time transformations. Every tetrad of magnitudes X L (t = l to 4) which are transformed as the x t , is called a four-vector or, after Min- kowski, a world-vector of the first kind. Such four-vectors are, in addition to dx t or x L itself, their prototype, the four- velocity dxjds and the four-acceleration of a moving particle, the electric four-current, and so on. To every vector X t LORENTZ TrANSFOKMTAIO F> belongs a scalar or invariant X\ l — X 2 — X-? — X£, its only invariant with respect to the Lorentz transformation. Bui we need not stop here to reconsider the properties of the four- vector and other world-vectors, such as the six-vector, which constituted the only lawful material of the older relativity for writing down laws of Nature, — especially as we shall soon return to these mathematical entities as particular cases of tensors of various ranks which are indispensable to the general theory of relativity. On the other hand we may profitably dwell yet a while upon the quadratic form (1) itself, the square of the line- element, as ds is called. Granted the assumptions of special relativity, this expression becomes the fundamental quadratic differential form of the four-manifold, the world or space - time, in exactly the same way as da 2 = dx 2 -\-dy 2 is the fundamental form of a flat two-space or surface, and more generally, da 2 = Edu 2 >-{-2Fdudv-\-Gdv 2 that of any surface, and d<r 2 =dr 2 +R 2 sin 2 ^ (sin 2 <£ d6 2 +d<j> 2 ) the fundamental differential form of any three-space of con- stant curvature i?~~.f Now, it is enough to open any book on differential geometry to see that, with the usual assump- tions of continuity, etc., the whole geometry, i.e., all metrical properties of the two-space or the three-space in question are completely determined by the corresponding differential forms. Their geodesies or, within restricted regions at least, their shortest lines, the angle relations, and their whole trigometry, all this is fully determined provided the co- efficients of the differentials, such as E, etc., appearing in fAccording as R 2 is positive, zero or negative, we have an elliptic, :lidean (or parabolic) or hyperbolic s becomes R sinh (r/R), where R 2 = — R 2 . euclidean (or parabolic) or hyperbolic space. In the latter case R sin - R 6 Relativity and Gravitation the fundamental form are given functions of the variables.* This deterministic mastery of the quadratic differential form has been, as far back as 1860, technically extended to spaces or manifolds of four and, in fact, of any number of dimensions, — although, not being sufficiently sensational, it never attracted the attention of anybody beyond a few specialists. In much the same way all the metrical properties of the four-dimensional world of the special relativist should be, and are, derivable from the fundamental form (1) belonging, or rather allotted to it. This is, from the point of view of the poly-dimensional differential geometer, but a very special, in fact, the most simple quadratic form in four variables. For it contains but the squares of their differentials, and the coefficients of these are all constant, which — in view of the sequel — it may be well to bring into evidence by writing (1) ds 2 = g LK dx L dx K , (la) to be summed over t, k = 1, 2, 3, 4, tabulating the co- efficients, thus -10 0-1 (16) 0-1 1 and calling this array of special coefficients the inertial or the galilean g lK . We shall denote them in the sequel by g lK . To give this array is as much as to give the form (la), and herewith the properties of the world, — for it is manifestly irrelevant how we call or denote the four corresponding variables. The values of the g lK being given, the properties *To be rigorous we should have said ' all properties of a restricted region of the contemplated manifold'; for certain properties of the manifold as a whole are still left free. The choice, however, is limited to a small number of discrete possibilities. Thus, for example, there are two kinds of elliptic space, the spherical or antipodal, and the polar or elliptic proper. In the former the total length of a straight line (geodesic) is 2tR, and in the latter irR; the planes are two-sided, and one-sided, respectively, and so on. Minimal Links AND GeODI « of the x will follow by themselves. There i no need to declare beforehand thai they are cartesian coordinates of a place and its date. Further, the circumstance that th< coefficients arc of different signs, three being negative, and one positive, creates for the general geometer no difficulty. This circumstance brings only with it the important feature that there are in the world real minimal lines* as the geometer would put it, that is to say, lines of zero-lenytli. ds = o. or dxi 2 + dx2*-\- dx^ dxi 1 c 2 \dis 1. These special world lines represent the propagation of light or, apart from physical difficulties, the uniform motion of a particle with light velocity c. As a matter of fact the very first step of the theory consisted in writing ds = as the ex- pression of light propagation in vacuo. In the next place consider the equally fundamental con- cept of the geodesies of the world. These are defined by 8/ds = o. the limits of the integral being kept fixed. To derive from this variational equation the differential equations of a geodesic, proceed in the well-known way. If u be any inde- pendent parameter, and if dots are used for the derivatives with respect to it, we have 8fsdu=J8s . du — o. where, by (1), s~= — (*i 2 +.^ 2 +*3 2 )+*4 2 > and therefore, 8 s = -r (x 4 5^4 — £i5xi— x 2 5.v" 2 — £35x3). 5 *Whereas on any (real) surface all the 'minimal lines' (known also as null-lines), which play in the surface theory an important analytical r61e, are always imaginary. The reader will do well to consult on this and allied topics a special treatise on differential geometry. 8 Relativity and Gravitation By partial integration, and remembering that all 8x t vanish at the limits of the integral, - xfix,, . du— — \ — ( — x t )8x t . du. s J du\ s / Thus, the 8x t being mutually independent, the required differential equations are ©=' d : : .0. du If the geodesic does not happen to be a null-line (light pro- pagation) we can as well take u = s, when s=l, and the equations become ds 2 whence Q/OC. (jsdC, Q/jO^ — = = a t = const. ds dxi ds The fourth of these equations is dxt/ds = const., and therefore, the first three, dxi _ dx 2 _ dx 3 _ — ^i> — °2> — o-z, dt dt dt and these represent uniform rectilinear motion, which is the motion of a free particle. Let us, therefore, keep well in mind these two properties of the line-element ds of special relativity: I. The minimal lines of the world, ds = 0, (I) represent light propagation in vacuo. II. The world geodesies, defined by 5fds = 0, (II) with fixed integral limts, represent the motion of a free particle. Minimal Lines and Gbodb 9 A special emphasis is here put on these two prop because they will be carried over to the general relativity and gravitation theory, and because these and principally only these two properties constitute the connection of the otherwise purely analytical differential form ds 2 = g^dx^dx, with physics. In other words, (I) as the equation of light propagation, and (II) as that of the motion of a free particle impart physical meaning to the mathematical form which is the 'line-element' ds. Without this all the properties of the quadratic form, though interesting, perhaps, in them- selves, would have nothing to do with the world of physical phenomena. It is scarcely necessary to say that the law (II) of the motion of free particles is, as well as (I) for light, invariant (thus far) with respect to the Lorentz transformation. For it is, by its very structure, independent of the choice of a reference system 5. Since ds is invariant, so \sfds, extended between any two world-points. Thus also the developed form of (II), the system of differential equations d 2 xjds- = 0, is transformed in S' into dhc l '/ds 2 = 0. And in fact, uniform motion of a particle relatively to S, means also (originally by an assumption) its uniform motion in any other inertial system S'. In short, the Lorentz transformation leaves the uniformity of motion of a particle intact. 5. We are now ready to pass to Einstein's theory of general relativity and gravitation. Not that our task is an easy one, but we are somewhat better prepared to embark upon it. Why equal form of physical laws, why equal rights for the inertial systems only? Why not equal rights for all (systems)? Such would be the urgent, and yet vague, ques- tions naturally suggesting themselves after what was said in the preceding sections. Yet it is not with these questions, nor with an attempt to answer them, that we will begin our journey across this new and revolutionary country. For, even if answered, these questions would remain physically barren were it not for the existence of gravitation and 10 Relativity and Gravitation especially of a certain peculiarly simple property of this universal agent. This, therefore, will first occupy our attention for a while. The cardinal feature of gravitation just hinted at is the pro- portionality of weight to mass, in other words, the proportion- ality of heavy (gravitating) and inert mass. First tested by Newton in his famous pendulum experiments with bobs of different material, and carried to further precision by Bessel, this proportionality has been more recently shown by Roland Eotvos to hold to one part in ten millions. It is reasonable, therefore, to assume, with Einstein, that it holds rigorously,* at least until proofs to the contrary are forthcoming. In our present connection it is better to express this property more directly by saying, even with Galileo, that all bodies, light or heavy, fall equally in vacuo. All particles, that is, acquire at a given place of a gravitational field equal accelerations independently of their own mass or chemical nature, etc., and no matter how much of their inertia is due to the energy stored in them and how much of other origin. This remark- able property distinguishes the gravitational field from other fields. Take, for instance, an electric field given by the vector E. The force on a particle of rest-mass m, carrying the electric charge e, and starting from rest, is eE, and the accelera- tion eE/m. Now, in general, there is no relation between m and e, and even if the mass is purely electromagnetic, when m is proportional to e 2 /a, the acceleration will vary from particle to particle inversely as its charge and directly as its average diameter, 2a. We have disregarded, of course, the dielectric properties of the particle which would make its behaviour in a given electric field still more complicated. The same remarks would hold, mutatis mutandis, for the behaviour of different bodies placed in a magnetic field. In short, gravita- tion is, in this respect, unique in its simplicity. *In a theory of matter and gravitation proposed by G. Mie, Annalen der Physik, vols. 37, 39, 40 (1912 and 1913), the proportionality between weight and mass does not hold rigorously, though to an order of precision much exceeding that stated by Eotvos. Equivalence Hypothj 11 This very circumstance enabled Einstein to undertake his mental experiment with the falling or ascending elevator, now so familiar to the general public. In fart, consider a homogeneous or a quasi-homogeneous gravitational field such as the terrestrial one in a properly restricted region. Lei a lift or elevator, small compared with the earth, yet ample enough for a physical laboratory and for those in charge of it, descend vertically with the local terrestrial acceleration g. Then all bodies placed anywhere within the elevator and left to themselves will float, in mid-air or better in vacuo, and particles projected in any direction will move uniformly in straight paths relatively to the elevator. Moreover, all objects, including the physicists, standing or lying about will cease to press against the floor or the tables, as the case may be. In short, all traces of gravitation will be gone,* and the inmates of the lift, assumed to have no intercourse whatever with the outer world, will declare their reference system to be a genuine inertial system, — so far, at least, as mechanical phenomena are concerned. For an unbiassed judge could not tell beforehand whether it will be also optically inertial, that is to say, whether the law of constant light velocity will hold good for the lift. Einstein thinks it will, or rather assumes it, more or less implicitly. If this be granted, we can say that the elevator will be an inertial reference system in every respect. The possibility of thus undoing a gravitational field is manifestly based on the said equal behaviour of all bodies placed in it. For otherwise the artificial motion of the elevator could not be adapted to all bodies at the same time, each of them requiring a different acceleration. Next, pass to any, non-homogeneous gravitation field, which in the most general case may also vary with time. This certainly cannot be undone, as a whole, by a single elevator as reference system. But you can imagine an ever increasing number of sufficiently small elevators, each appropriately accelerated, fitted into small regions of the field, and each, *Vice versa, in absence of a gravitational field, a lift in accelerated ascending motion would give us a faithful imitation of such a field. —2 12 Relativity and Gravitation perhaps, to do its duty for a very short interval of time, and to be replaced by another in the next moment. These minute elevators will do their office at least in the mechanical sense of the word. Einstein assumes that they will act as inertial systems also in the optical sense of the word, as explained above. This process of subdividing a gravitational field, in space and time, and fitting in of appropriate small elevators can be carried on to any required degree of approximation. In fine, passing to the limit, let us make, with Einstein,* the explicit assumption: With an appropriate choice of a local reference system (ui, u 2 , u z , Ui) special relativity holds for every infinitesimal four-dimensional domain or volume-element of the world. That is to say, at every world-pointf a system of space- time coordinates u x , u 2 , u s , u± can be chosen in which the line- element assumes the galilean form ds 2 = du£ — du\—du 2 2 — du£. (3) These four coordinates are called local coordinates. With respect to this local system there is then no gravitational field at the given world-point, and in accordance with special relativity ds 2 has there a value independent of the 'orienta- tion ' of the local axes; that is to say, the quadratic form (3) is invariant with respect to the Lorentz transformation (2). It is this assumption which can now be properly referred to as the infinitesimal equivalence hypothesis, for it grew out of Einstein's original equivalence hypothesis applied to finite regions when, in his first attempt at a theory of gravitation (1911), he was confining himself to a homogeneous field. Whatever the origin of this hypothesis or assumption, it is certainly not difficult to adhere to it. For it scarcely amounts to anything more than to assuming, in the case of a curved surface, say, the existence of a tangential plane at any of its *A. Einstein, Die Grundlagen der allgemeinen Relativitatstheorie . Annalen der Physik, vol. 49, 1916, p. 777. fWith the possible exception of some discrete points, such perhaps as those at which the density of matter acquires enormous values. Infinitesimal World Flatni i 3 points, or to <lc( lare the surface to be (in Clifford*! termino- logy) elementally flat. And it will, perhaps, I"- wt 11 to shortly Einstein's hypothesis by Baying thai it the jour-dimensional "world to he, in presence as well a- in ab • of gravitation, elementally /Jul. It will not be forgotten, I over, that this geometric term is nothing more than a synonym of elementally galilean, i.e., satisfying special relativity in- finitesimally.* To avoid the danger of any misconception let u- dwell upon this subject yet for a while. The coordinates u t with their corresponding galilean line-element (3) were set up only for a local purpose, their real office being confined to a fixed world-point P, say x u x 2 , x 3 , x* (in any coordinate system). If we so desire, we may think of a whole galilean world U determined throughout, to any extent, by the simple form (3). But as a tangential plane has something in common with a surface only at the point of contact and then diverges from it, ceasing to represent any intrinsic properties of the surface itself, so has the auxiliary and fictitious world I' anything to do with the actual world W (complicated by gravitation) at the world point x t only. The fictitious world Uis tangential to the actual world IF at that point, and parts company with it beyond the point of contact. At other world-points the role of U is taken over by other and other fictitious galilean worlds. One more cautious remark. The contact of U and W is one of the first order, i.e., such as the contact between a surface and its tangential plane or between a curve and its tangential line, but not as the more intimate contact bewteen a curve and its circle of curvature (which is of the second order). This circumstance may acquire some importance later on. *As to the concept of elementary flatness of a surface or a more-dimen- sional space, it is beautifully explained in W. K. Clifford's ' Philosophy of Pure Sciences', published in his famous Lectures and Essays (Macmillan, London). Notice that in Clifford's sense every regular surface, no matter how curved, is elementally flat, with the exception of some singular points, such as the vertex of a cone. 14 Relativity and Gravitation 6. Having thus made clear the local character of the w t coordinates, let us now introduce any coordinate system x 4 whatever, to be used as a reference system of coordinates for the whole world, i.e., throughout the gravitational field and through all times. Then, if x t be the reference coordinates of P, and x t -\- dXt those of a neighbour-point u t -\-du lt the differentials du l will in general be linear homogeneous func- tions of all the dx L , say 4 (LU L —- 2j CL lK (tX K , or with the conventional abbreviation, du t = a lK dx K , (4) where the coefficients a lK will in general be functions of all the x t . It is of importance to note that that the relations (4) will, generally speaking, be not-integrable, or borrowing a name from dynamics, non-holonomous, that is to say, the a lK will not necessarily be duJdx K , the differential expressions on the right of (4) will not be total differentials of functions of the x t , and there will be no finite relations between the local and the general or reference-coordinates. Substituting (4) into (3), collecting the terms and calling g lK the coefficient of the product of differentials dx t dx K we shall have, for the line-element in the general reference system, ds 2 = g tK dx t dx K , (5) where g lK = g Kl will, in the most general case, be functions of all the x t . But since ds 2 , as defined originally by (3), was in- dependent of the orientation of the local system of axes, so also will the ten different coefficients g lK , though functions of the coordinates x t , be manifestly independent of the orienta- tion of the local system. The line-element will thus be represented, in any reference coordinates x t whatever by the most general quadratic differen- tial form of these four variables, such in fact being the form (5). As before the summation sign is omitted; the sum- mation is to be extended over i, k from 1 to 4, each of these Cknkkal Tkansfomiati 15 suffixes, l and k, appearing twice. Thus, ds 2 = i[ndx l *+ 2g u dx\dxz-\- . . . +f>udxi 2 . The reader will soon learn to handle this abbreviated and very convenient symbolism. Suppose now we introrluce instead of X t any other space-time coordinates x/, any functions whatever "I il i such, that is, that between the two sets exist any given holo- nomous relations *i=&(*i', x 2 ', x 3 \ xS), (6) the <f> t being any functions whatever, but continuous together with their first derivatives and such that their Jacobian, the well-known determinant J dx t dxj dx\ dxi dx\ dx 2 ' dx\ dXi dx 4 dXi dXi dx\ (7) dxi' dxo' does not vanish. Under these circumstances we have dx t = — -dx K ' ', dxj and vice versa, dxj dx/ = — - dx K , dx r (8) (8a) and, as it may be well to notice in passing, //'= 1, where /' dx{ is the inverse Jacobian dx M Now, substituting (8) into (5), gathering again the terms, and denoting the coefficient of dx t 'dx K ' by gj , i.e., putting , dXa chj, dx/ d.v, - &afi, we shall have ds 2 = g LK f dx l 'dx K ' (9) (50 which is (5) reproduced in dashed letters. Not that the gj will be functions of the x/ of the same form as were the g^ of 16 Relativity and Gravitation x„ but only that the quadratic differential form remains quadratic. There is certainly nothing surprising in this kind of permanence.* Yet this and this only is justifiably meant when we say that the line-element ds 2 is invariant with respect to any transformations whatever. If the relativist sees anything more in "the invariance of ds", namely that ds is something belonging to a pair of world-points (x, and x t -\-dx t ) inherent in that pair independently of the choice of a reference system, it is what he puts into it at a later stage by ascribing to it certain physical properties, or by inter- preting it physically in certain ways. The meaning of these remarks will gradually become more intelligible. Before passing on to the two cardinal virtues conferred upon the line-element, one more mathematical remark about it may not be out of place just here. Suppose the line- element (5) is actually given with some determined and more or less complicated functions as the g lK . By trying, in succes- sion, other and other new variables xj we would arrive at a great variety of new forms of functions gj . The natural question arises: Are there not among all these sets of co- ordinates just such as would convert (5), throughout the world or a finite world-domain, into a galilean line-element, i.e., one with constant coefficients? The answer is, in general, in the negative. A given form ds 2 = g lK dx t dx K is equivalent, that is to say, can be reduced by holonomous transformations, to a form with constant coefficients and thus also to the galilean line-elementf when and only when certain differential expressions formed of the g^, their first and second deriva- tives, all vanish.* These expressions, of which more will be *Notice that the case is different in special relativity, where we require the form to reappear with all its original coefficients, three — 1, and one+1. fThe circumstance that three of the coefficients of this form are negative and one positive imposes on the original g dx dx to be thus transformable certain further conditions in connection with the so-called 'law of (alge- braic) inertia', due to Sylvester. *The restriction to ' holonomous transformations ' is of prime importance. For by means of non-holonomous or non-integrable relations, such as (3), every g dx t dx can be transformed into a quadratic differential form with constant coefficients. Rjemann's Symbol it said in the sequel, arc known in general differential geometry as Kiemann's four-index symbol*. Of ' symbols there arc in the ease of any number n of din — n 2 (n 2 — 1) linearly Independent ones. Thus an ordinary, two-dimensional, surface has hut one Riemann symbol and this is its Gaussian curvature, multiplied by gng22 — ga* t tin- determinant of the g, K . Any three-dimensional manifold has six, and our, or rather Einstein-Minkowski's world has as many as twenty linearly independent Riemann symbols. Thus any finite domain of the world is equivalent to a galilcan domain when and only when all these twenty symbols vanish in that domain, i.e., when the ten different £« satisfy within it a system of twenty partial differential equations of the second order. (It will be useful to keep in mind the last italics.) By what has just been said it is manifest that if all the Rie- mann symbols vanish in one system of coordinates x t , they will vanish also in any other %/ obtained from the former by any holonomous transformations whatever. But enough has for the present been said on the symbols of that great geometer. Later on they will be seen to play an all-important role in Einstein's gravitation theory. It is now time to return to the physical aspect of our subject. 7. Having assumed, after Einstein, that special relativity holds for every infinitesimal domain, or that the world is elementally galilean, we wrote down the simple form (3) in local coordinates u t . Then, passing to any coordinates x t by means of the non-holonomous relations (4) w r e obtained for the line-element of the world the general quadratic differential form (5), with variable coefficients g lK , functions of the x g . But what is the physical meaning of this general ds with all its ten different g lK ? What are they to represent physically ? The answer is that we are still to a certain extent the masters of the situation, and can make them have that physical meaning which we will put into them. For thus far we know only the physical meaning of the galilean element belonging to a world U, and that (in virtue of an assumption^ the world 18 Relativity and Gravitation W as a seat of or deformed by gravitation is galilean in its elements, or that at each of its points a £/"-world tangential to it can be constructed. At this stage then we are entitled only to say that (since ^without gravitation is If, and since to If belong the constant coefficients g tK ) the essential differences* in the coefficients of the two worlds, g lK and ~g lK , are due to, or better, are somehow connected with gravitation. But exactly how, we cannot, thus far, say. For our position is somewhat like this : Suppose we know that a surface a, which is not a plane as a whole, is elementally flat and thus has a tangential plane tt at each of its points. Suppose further we know the physical properties of certain lines (straights, or circles, etc.) drawn on any ic. Does this alone enable us to say what the physical properties of similarly defined lines will be when drawn on o-? Clearly not. For the 7r-lines have but a single point of contact with a, and that only of the first order, and deviate from the surface or become extra-a beings all around the point of contact. Now, in the case of space-time, we fixed the physical meaning of the line-element of the LV-world by declaring its minimal lines, ds = 0, to be the law of light propagation, and its geodesies, 8fds = 0, to represent the motion of free particles. Does this, and the existence of a tangential If at every point of the actual world W, entitle us to assert that the minimal lines and the geodesies of W will again represent the optical and the mechanical laws in this world? This is by no means a superfluous question. For the auxiliary tangential world U leaves the actual world beyond the point of contact and becomes at once fictitious or extra-mundane, so to speak. Now, the minimal lines of t/,t defined by a differential equation of the first order, are also, at P, minimal lines of W, so that at least the starting elements of these lines are identical. At the next element the r61e of If is taken over by another *i.e. t those, at least, which cannot be abolished by holonomous co- ordinate transformations. fWhich fill out only a conic hypersurface (of three dimensions) with the contact point P as apex. GEODESICS AM) Law 01 Mom- l'» galilean world; yei the reasoning can be repeated, jo thai can say that every elemeiii of ;i minimal line "I W represents light propagation, and thence deduce that such a ll'-line possesses also as a whole the same physical property. Fait the position is altogether different with the geodesil I or these world-lines are defined by differential equations of the second order* so that the mere contact of U and W (bein the first order) does not at all entitle us to transfer any properties of the geodesies of U upon those of W, not even at their very starting point P. If, however, the said physical property of the K'-geodesics does not follow logically from the previous assumptions, yet we are free to introduce it as a further explicit assumption. In fact, while thus generalizing the physical significance of the geodesies Einstein is well aware that this is a new assumption,! although one that easily suggests itself. Nor is there any inconsistency in thus transfering a property from the galilean to the more general world-geodesies. For, as we shall see later on, the developed form of the equations of the geodesies contains only the g lK and their first derivatives with respect to the x„ whereas the conditions characterizing a world as galilean (the vanishing of the Riemann symbols) are equations between the g lK , their first and second derivatives, and there are no relations at all between the g lK and their first derivatives alone. But even with this new assumption, the total number of assumptions of Einstein's theory is remarkably small. And as to the advisability of making the one just discussed, we may say that Einstein's theory owes to it the greater part of its power. The property of the geodesies being thus assumed, and that belonging to the minimal lines being deducible from what preceded, we are now in the position to sum up definitely and very concisely, if not the whole, yet the most fundamental part of Einstein's theory. For this purpose we have only to *A geodesic issues from P in every direction whatever in the four- manifolds U and W. fA. Einstein, loc. ciL, p. 802. 20 Relativity and Gravitation repeat the previous statements I. and II. without their restrictions, replacing the galilean ds by the general one and adding a few explanatory words. Thus: The world-line element, in any system of coordinates, and whether gravitation be absent or present, is given by ds 2 = g lK dx t dx K , (10) where g tK = g Kt are some functions of, in general, all the jour coordinates, but of these alone. If these ten functions be given, all metrical properties* of the world are determined, and among these its minimal lines, ds = 0, (I) and its geodesies, 8fds = 0. (II) The physical significance of these world-lines is that the former represent propagation of light in vacuo, and the latter the motion of a free particle. By a 'free' particle is meant one which, having received any initial impulse is left to its own fate, whether in absence or in proximity of other lumps of matter (absence or presence of 'gravitation'), but not colliding with them, and in absence of, or better not immersed in, an electromagnetic field. One strives in vain to enumerate all the attributes of a concept which can become clear only a posteriori, through the concrete applications of the theory. Suffice it to say that ' free particle ' may as well stand for a projectile, in vacuo, or a planet circling around the sun. Their laws of motion are given by the corresponding world-geodesies. The developed form of the equations of the geodesies, as well as of light propagation, will be given later on. Since the g lK are to determine, through (II), the fall of projectiles and the motion of celestial bodies, it is scarcely necessary to repeat that they are intimately connected with gravitation. These ten coefficients will replace the unique scalar potential of newtonian mechanics. They will influence * Apart from some properties of the world as a whole, — of which more later on. Free Particles AND LlGHl J I also, through (I), the course of light in interplanetary ar.fi interstellar spaces, and finally, by t heir very appearance in the line-element, they will mould the geo- and ehrono-metrical properties of our world. These latter properties thus appear intimately entangled with gravitation and optics. It remains to explain how these all-powerful coefficients arc, in their turn, determined in terms of other things such as the density of 'matter'. This is the office of Einstein's 'field-equations' which will occupy our attention in the sequel. CHAPTER II. The General Relativity Principle. Minimal Lines and Geodesies. Examples. Newton's Equations of Motion as an Approximation. 8. Most readers will perhaps be surprised to find in the first chapter almost no mention of the general principle of relativity which claims equal rights for all systems of co- ordinates, and which in all publications on our subject is given the most prominent place. Instead of this we insisted on the general form of the line-element (10), on the null-lines and the geodesies of the world metrically determined by that line-element, and still more upon the physical meaning of these two kinds of world-lines as representing light propa- gation and the motion of free particles. The reason for adopting this plan is that, as far as I can see, these things are most important from the physical point of view, nay, they are perhaps* the only relevant constituents of the new theory looked upon as a physical theory. This is particularly true of the optical and mechanical meaning attributed to the said two kinds of lines, thus giving what the logicians call a concrete representation of what otherwise would be only a purely mathematical or logical science, an abstract geometry of a manifold of four dimensions deter- mined by that quadratic differential form. It is exactly this physical interpretation which invests the theory with the power of making statements of a phenomenal content, of predicting the course of observable events. On the other hand, the much extolled Principle of General Relativity which, in Einstein's wording,! requires The general laws of Nature to be expressed by equations valid *Apart from ' the field equations ', yet to come. \Loc. cit., p. 776. 22 General RELATIvm PRINCIPLE 23 in all coordinate systems, i.e., covariant with respect to any substitutions whatever (generally covariant), is by itself powerless either to predi< I or to exi lude anything which has a phenomenal content. For whatever we already know or will learn to know ahout the ways of Nature, pro- vided always it has some phenomenal contents (and is not a merely formal proposition), should always be expressible in a manner independent of the auxiliaries used for its descrip- tion. In other words, the mere requirement of general covariance does not exclude any phenomena or any laws of Nature, but only certain ways of expressing them. It does not at all prescribe the course of Nature but the form of the laws constructed by the naturalist (mathematical physicist or astronomer) who is about to describe it. The fact that some phenomenal qualities are technically (with our inherited mathematical apparatus) much more difficult to put into a generally covariant form than some others does not in the least change the position. To make my meaning plain, let us take the case of plane- tary motion. For the sake of simplicity let there be but a single planet revolving around the sun. It is well-known that according to Newton the orbit of the planet should be a conic section, say an ellipse with fixed perihelion.* It is, in our days, almost equally well known that according to Einstein's theory the perihelion should move, progressively, showing a shift at the completion of each of its periods. And so it does, at least to judge from Mercury's behaviour. At the same time Einstein's equations are generally covariant, while Newton's 'law' or Laplace-Poisson's equations are not.f What of this? Does it mean that fixed perihelia are excluded or prohibited by the principle of general covariance? Cer- tainly not. Provided that 'fixed perihelion' and 'moving perihelion' have, each, a phenomenal content, and this they do, both kinds of planetary behaviour should be expressible in a generally covariant form. Newton's inverse square law and his equations of motion certainly do not express it so, *Fixed, that is, relatively to the stars. fNot even with respect to the special or the Lorentz transformation. 24 Relativity and Gravitation and it may be difficult to find a covariant expression for a strictly keplerian behaviour. But if it were urgently needed, some powerful mathematician would, no doubt, succeed in constructing it. If, as actually is the case, Einstein's theory excludes a fixed perihelion, and other newtonian features, it does this not in virtue of the said principle alone (nor even in part), but pre-eminently owing to the physical meaning ascribed to the world-geodesies, and to the choice of his field equations which again are physically relevant since they determine the g lK influencing essentially the form of those world-lines. That the principle of general relativity turned out to be helpful in guessing new laws (by limiting the choice of formulae) is an altogether different matter. It may prove an even more successful guide in the future. f But here its role ends, — always taking the Principle only as a mathematical requirement of general covariance of equations. And so it is at any rate enunciated (and interpreted, cf, p. 776, loc. cit.) by Einstein himself, although some of his exponents put into it a physical meaning. In fact, as we shall see later on, the sameness of form of the equations (of motion, say) in two reference systems, as in a smoothly rolling and a vehemently jerked car, does not at all mean sameness of phenomenal behaviour for the passengers of these two vehicles. So much in explanation of the absence of the general principle of relativity in all our preceding deductions. It will be noticed, however, that although no explicit mention of this principle has been made in Chapter I, yet the fundamental laws (I) and (II) there given do satisfy this principle. In fact, both the null-lines and the geodesies of the world were defined without the aid of any reference system. And as to the line-element itself, its invariance was seen to be automatic. Thus, in what precedes we have, without insisting upon it, been faithful to the formal principle of general relativity. Nor is it our intention to depart from it in what will follow. fOr it may become sterile to-morrow, as is the fate of almost all our Principles. Light Veixx n v As was already mentioned at the close o( the first chapter, to make the expo iiion of the fundamental pan of Einstein's iheory complete, il remains to add to MO;, ( I J, (I I ), together with their optical and mechanical meaning, a -' I of equal ion - determining the ten coefficients #„ of the quadratic form. But before passing to these differential (filiation-, Ein t< in'- field-equations, it will be well to discuss somewhat more and to develop those already given. Some explanations and examples concerning the transformation of coordinates will also be helpful at this stage. 9. First, concerning the law of propagation of light (in vacuo), to obtain its developed form it is enough to sub- stitute the line-element (10) into the equation (I) of the minimal lines. Thus the fundamental optical law will be g lK dx l dx K = 0. (11) It gives the velocity of light for every direction of the ray, i.e., of the infinitesimal space-vector dx\, dx 2 , dx 3 , if dx\jc be the time element of the reference system. In general the light velocity will differ from c and have different values at different world points and for different directions of the ray. This "light velocity" which has nothing intrinsic about it is to be distinguished from the local velocity of light (that corresponding to a local, galilean system of coordinates) which is the same for all directions. To avoid confusion the former may be called the system-velocity of light or, according to some authors, the 'coordinate velocity' of light. It is a kind of velocity estimated from a distant standpoint. If we write it, in a given reference system, dcr _ da dt dXi ' the very concept of such a light velocity, whose value is to be derived from (11), presupposes that 'the length' da of the infinitesimal space-vector dx u dxi, dx* has been defined in some way for that system in terms of these differentials and the coefficients g lK . We shall have the best opportunity of 26 Relativity and Gravitation explaining how this is done technically in deducing physical results, when we come to speak of the bending of rays of light around a massive body such as the sun. Then also the question will be mentioned under what circumstances the law of Fermat, giving the shape of the rays, is applicable. In the meantime it is advisable to look upon (11) as the equation of the infinitesimal wave surface at the instant t-\-dt corresponding to a light disturbance started at xi, x 2 , x s at the instant /, the differential dx± being treated as a constant parameter. From the local standpoint this surface is, of course, a sphere, but from the distant (or system-) standpoint it may have a variety of more complicated shapes. It would, perhaps, be rash to say that it will be a quadric. But, being locally closed, it may also be expected to be a closed surface from the system-point of view. 10. Next for the geodesies of the world. The developed form of their differential equations is easily derived from their original definition (II), 8fds = 0. As in the case of a galilean world, let u be any parameter, and let dots stand for derivatives with respect to it. Then f8s . du = 0, where, in the most general case, S 2 = gucXtX K . (12) The variation of s can be written ds « i <?$ „. os = 5x t + — dX t , dx c dx t to be summed over t= 1 to 4. Thus, by partial integration of the second terms, the limits of the integral being fixed, d / ds \ _ ds_ - 0, du V dx. s dx. du \ dx L / dx t and by (12), with 5 itself taken for u, Geodes h rl:\ tK ds / " dx. ds (Is or ds 2 dx x ds <ls dx t ds ds Introducing the expressions, known as Christofft l's syml r«/n m l /a^ a d^ii _ ***) = p G l LtJ V^ dx a dxy/ LtJ we can condense the last set of equations into a ^5 -u |~ a,3 ~l ~" — = n gut ds^L^Jds ds ' These are four linear equations for the four d-x K , ds"-. Let us solve them for these derivatives. Denoting the second term by a„ and writing g for the determinant of the g„, we shall have d 2 Xi 1 ds 2 g fll g]2 gl3 gU a 4 g42 gi3 gii = 0, etc. or, if g uc = g'" be the minor of g, corresponding to g^, divided by g itself, — -r + «ig u + Ofig 12 + a 3 g 13 + a^ 14 = 0, etc., ds- t.e., </s ? * L « J ds ds Here we will write, after Christoffel, -3 fa/3) o „ra/T| .. )o'a\ (14) 28 Relativity and Gravitation Thus, ultimately, the differential equations of the geodesies or the equations of motion of a free particle will be, in any system of coordinates, (^ + {"P\dx« ^% =0 * (1 5) ds 2 \ >■ ) ds ds These are four equations. But since we have, identically, ds as one of these equations of motion is a consequence of the remaining three, a feature already familiar to the reader from special relativistic mechanics. Since these differential equa- tions are only the developed form of 8fds = 0, they will mani- festly be generally covariant, that is to say, in any new coordinates xj the equations (15) will be d 2 xj jap\'dx a ' dx/ _ ds 2 \ L ) ds ds If the coefficients g lK are all constant, all the Christoffel symbols < > vanish and the equations (15) reduce to d 2 xjds ? = 0, which represent uniform rectilinear motion. And since the general equations (15) represent the motion of a free particle in any gravitational field and in any system, the symbols < V, built up of the g lK and their first derivatives, can be said to express the deviation of the motion from uniformity due to gravitation, and partly due to the peculiari- ties of the system of reference. In view of this property, and disregarding any distinction between gravitation proper and the effects of the choice of the coordinate system, f Einstein *This form of the equations of a geodesic of a manifold, of any number of dimensions, has been used by geometers for a long time. See, for instance, L. Bianchi's Geometria differentiate, vol. I, Pisa 1902, p. 334. fOr between permanent acceleration fields and such that can be trans- formed away. Christoffel Symboi proposes to call these ( Ihristoffel symbols 'the components of the gravitational field '. Notice, however, that if all \ ( vanish in om 1 ' ) of reference they do not necessarily vanish in other systi (even if obtained from the former by holonomous transforma- tions). In view of this circumstance the name proposed by Einstein seems utterly inappropriate and misleading, even if one agreed not to distinguish between permanent fields and such that can holonomously be transformed away, as for instance the 'centrifugal force'. 10a. In fact, consider for example the galilean line-element in three dimensions, i.e., for <f> = const. = t/2, ds~ = c 2 dt 2 — dr~ — r-dd' 1 , taking ct, r, d as x 4 , Xi, x 2 respectively. Calculate the corres- ponding Christoffel symbols. Since gn= — 1, gja = — r 2 , g« = 1 , and all other g lK vanish, we have, for instance, the non-vanish- ing symbol i22l 1 But who would call it a 'component of the gravitational field '? This case is a particularly drastic one, for the world-geodesies corresponding to our line-element do represent uniform recti- linear motion. The appearance of non-vanishing Christoffel symbols is simply due to the use of polar instead of cartesian co-ordinates. In short, gravitation certainly contributes to the Chris- toffel symbols, but so does also a mere transformation oi space-coordinates, although it. has nothing whatever in common with 'gravitation' of the permanent or the non- permanent kind. This criticism does not in the least diminish the value of the general equations of motion (15). It is given here only to prevent misconceptions which have seemed particularly likely in the case of beginners. *In the terminology of the tensor calculus, to be explained later on, the Christoffel symbols are not the components of a tensor. 30 Relativity and Gravitation 10b. Let us take yet another simple example, this time not for the sake of criticism but because of its instructiveness. Consider the line-element arising' from the galilean one, just quoted, (S') ds 2 = dx' i 2 -dr' 2 -r' 2 dd' 2 , by the transformation 6' = 6+00x4, x'i = Xi, r' — r, (16) that is to say, the line-element (S) ds 2 =(l- r 2 <a 2 )dx^ - dr 2 - rW - 2cor 2 d6dx i . In this case, taking r, 6 as Xi, x 2 respectively, the non-vanishing gtK are g 11 = - 1 , £22 = - r 2 , g 24 = - wr 2 , g u = 1 - co V. From these we derive, by (13), as the only surviving Chris- toffel symbols, 0)% r22~i r24~i r44"i , Li_r r ' lij =0 "'±ir ar - Next we have, the determinant of the form (S), g = gn(g22gu-g2i 2 )=r 2 , and arr 2 — 1 gii=_l 7g2 2 = ' t g24==a;>g 44 =1> r 2 while all other g lK vanish. Thus we find, by (14), as the only non-vanishing symbols, 12) l-2co 2 r 2 (12) (14) = -2cor, < > = — (l-2orr-), ftl 2 r 4 — '2 141 n „ 22) (44) • (24 l }~*- {?}—{?} 4 j " w '' (1 J~ '' (l j = -arr > 1 1 r = again seven in number. Substituting these ChristofTel symbols into (15), with i = l, 2, 4 (for r, 6, Xi = ct), we have the equations of the world-geodesies, i.e., the equations of motion of a free particle in the system S, Rotating Sysi em 31 r =r (d+oiXiY =-— (l-2co 2 r 2 )(0 + o;.-v,) (17) X4 = 4o)r.r (0 +o)X 4 ), where the dots stand for derivatives with respect to s. In virtue of the identical equation 5 = 1, i.e., ( 1 - r 2 w 2 )x 4 2 - r 2 — r 2 6 2 - 2wr 2 dXi = 1 , (18) one, say the third of (17), should be a consequence of the remaining two.* Thus, the proper equations of motion in theS- system being the first two alone, we can use (18) to eliminate from them x A , and to replace d/ds by d/dt. In the first place, to see the approximate meaning of these equations of motion, consider the case of small velocities dr/dt, rdd/dt (as compared with c), and of small values of ur. [Notice that, by (16), « = a>c is an angular velocity, in its dimensions at least, so that ur= 6or/c is a pure number.] Thus ds=¥dx4 = cdt, £ 4 =f1, and the approximate equations of motion of a free particle in 5 are dr 2 dd_ dt dt 2 d 2 6 dt 2 -(*+■&. (a) = — 2 dr dt (*+-£) In Cartesians, identical with x = rcos6, y = rsin0, these equations are d 2 x ~dF (Py_ It 2 -2 V i o - d y dt . dx dt (b) The reader will recognize at once in the right hand member of equation (a) or in the first terms of (b) the purely radial centrifugal acceleration (or 'force' per unit mass), provided, *The verification may be left to the reader as an exercise. 32 Relativity and Gravitation of course, that he is at all willing to interpret o5, in accordance with the transformation d' = d-\-u>t, as the angular velocity of the system 5 (say, plane disc) relatively to the galilean S'. The second terms of (b) express then the Coriolis acceleration. If we so desire we may, with Einstein, reckon these accelerations to the gravitational ones, especially if we are confined to the (rotating) system S. The centrifugal acceleration, at least, is radial, though away from the origin. The Coriolis acceleration, however, is perpendicular to the velocity and, therefore, generally oblique. Certainly we have in (17) a field of acceleration, but the only feature this has in common with a gravita- tional field is that all bodies placed in it will behave alike. But unlike gravitational fields they cannot be deduced from the distribution of matter. Yet Einstein would not like to have us distinguish them from gravita- /22( J24I 1441 tional fields. If so, then \ , ( , ) i ( > i i ( contribute to the centrifugal, and < " > , \ > to the Coriolis field. But until we are told how to derive these non-permanent 'fields' as gravitational effects of all the masses of the universe turning around S* all this will be an idle question of pure nomenclature. We may leave it here for the present. In the second place, returning to the rigorous equations (17), consider a particle, placed (by an 5-inhabitant) at any point r , 8 of the disc 5 and left there, at the instant t = 0, to its own fate. If it is nailed down it will, of course, remain there for ever, being simply part of this reference system. But let it be a free particle from t = onwards. In short, let r = = 0, for / = 0. Then, by (17), we shall have, for that instant, 6 = so that the particle will not evince any tendency of moving transversally, and d*r . _d_ ds 2 dxi (. dr\ 5 V dt / By (18), x 4 2 = (1 — r 2 co 2 ) , and since r o = 0, the last equation will become, rigorously, and always for / = 0, d 2 r .„ =co 2 r. dt 2 *This was tried by H. Thirring but not very s uccessfully. Rotating S\ stem 33 In fine, our particle will initially experience tin- familiar centrifugal acceleration.* It will fly off, for an V-ob-erver at a (straight) tangent, but from the S-standpoint at a spiral-shaped orbit. This is perhaps the clearest way of Btating the relation of our system S to the galilean S'. The reader need noi , however, think of 5 at this stage as a material rigid dia rotating uniformly with respect to the fixed stars, although a uniform rotation is just one of the possible motions of a relativistically rigid body (Born, Herglotz). Notwithstanding that 5 was called, in passing, a disc, it will be safer to treat it here simply as a system derived from S' by the trans- formation (16) with aj as constant. As to the orbit of a free particle relatively to S, its equation could be derived, not without some trouble, from the differ- ential equations (17). This, however, can be done much easier by transforming the orbit from S' to S. In fact, the former being a galilean system, a free particle describes in it, uniformly, a straight line. Its equation can be written r' cos 8' = r f = const., where r ' is the shortest distance of the straight orbit from the origin. Transformed by (16) the orbit in 5 will be 12 = cos (6+5>t'), r and since v't'= V r- — r 2 , where v' is the constant S'-velocity of the particle, we shall have ultimately, as the orbit of a free particle in S, ^ = cos [ e± %° v / ZT,~\. (19) r L v v r 2 _l *One of Einstein's most vigorous exponents, de Sitter, sees herein a particularly extravagant property of the rotating system. Thus in Monthly Notices of the Roy. Astron. Soc, vol. 77 (1916), p. 176, de Sitter says: 'For rco<l' [and, as we saw, for any ro>] 'it is a physical impossibility for a material body to be at rest in the system B' [our S\. 'This shows the irreality of the coordinates', etc. But such is, in reality, the behaviour of free particles in a system rotating relatively to the stars, independently of any theory. 34 Relativity and Gravitation which is a kind of spiral. Notice in passing that between any two points A, B of the disc there are two such orbits, one leading from A to B and the other from B to A . Thus free motion in 5 is not reversible. This holds also for light rays, for which v' in (19) is to be given the value c. Light propaga- tion is irreversible, and the two rays AB and BA enclose a certain area having the shape of a biconvex lens. But this by the way only. The example of these two systems, S' and S, was here treated at some length in order to acquaint the reader with the handling of the geodesies and the Christoffel symbols. At the same time, however, it may serve as a good illustration of the purely formal part played by the principle of general relativity or general covariance. In fact, although the equa- tions of motion of free particles have exactly the same form, (15) and (15) dashed, in the two systems, yet it is scarcely possible to imagine a more different phenomenal behaviour of free particles than is that in these two systems. The same remark applies to the light equations, gj dxj dxj = in S' and g uc dx l dx K = in S, exhibiting the same general form, but representing entirely different systems of optics; this differ- ence goes even so far that, while in S' all light paths are reversible, in S, under appropriate conditions, Brown could see Jones without being visible to him, though both were well enough illuminated. The purpose of these remarks is by no means to minimize the heuristic value of the general relativity principle, but only to show its purely formal nature. Notice that the case of the special relativity theory was altogether different; for, though giving privileges only to a certain class of systems, it claimed at least for all of them not only a formal equality, but an equal physical behaviour. In passing from S' to 5 the Lorentz contraction was, for the sake of simplicity, altogether disregarded. This is the reason why the reader was warned not to take our .S strictly as a rigid body rotating in S' but only as one obtained from S' by the simple mathematical transformation (16). Yet even with the said neglect the abstract 5 can at least Kni \ [IONS oi Monoi approximately stand for a rigid body, such as the <-.trtli plane r, parallel to its equator), endowed with a uniform spin relatively to the stars. 11. Leaving these simple examples let us once more return to the genera] (filiations of motion of a free particle, in order to see what form they assume when the g iK differ bill little from the galilean coefficients #« and when x u .r_>, .v 3 are small fractions, that is to say, when the velocity of the particle is small compared with that of light. If the galilean line-element is written in Cartesians we have gii = g 2 2 = g33= -1, #44=1, and gn= - 1 + Yn, etc., g 4 4= 1+74-i, 1 / 21 x where all the 7 are small fractions of unity. With these values of the g^ we could compute the approximate values of the Christoffel symbols appearing in (15), and thus arrive at the required equations. But it is simpler to return to 8fds = 0, the original form of (15), to reduce the element ds and then to develop this form afresh. Now, if dxi/cdt = f3 u etc., i8rH-/3 2 2 +|83 2 = j8' 2 , the line-element can be written ^ 2 = </* 4 2 {l-/3 2 + 744-(7n& 2 + • • • +733/3 3 2 )+2(7 12 i8i&+ . . . + 73103|8l)+2(7l4i8l+ • • . +734&)}. All the squares and products of the j8's are small of the second order. Thus, up to the third order we have ds = Ldx i = dx 4 Vl-p + lu + 2{p l y u + . . . +0sYm), (22) and the equations of motion, JSL . <:/.v 4 = 0, will be d ±(dL\_BL =Qi=lt 2,3. d: ' 36 Relativity and Gravitation L--J_C -R) dL - 1 T 1 dyu +B djil 4- Jf L dx t - L L-2 dX{ dX{ —I Now, and if the squares and the products of the 7's and their derivatives be neglected, we can putZ,=l in the denominators. Thus the equations of motion will become d , o\ -i ^744 , a d74i , a dy i2 . a dy i3 (7«- ft) = h • r-Pi ■ +/3 2 \~Pz dxi dxi dxi dxi dXi or, developing the first term and remembering that the y L differ from the g lK only by additive constants, d 2 Xi IF C 2 dg u , r dgq dxi a 2 dxi L dt dt ' , ( ^i _ dg43 V dx 3 dXi /dgu _ dgu \ \ dxi dXi ' )]■ + ••• (23) These are Newton's equations of motion. The first terms on the right hand represent the rectangular components of an acceleration which is the gradient of a newtonian potential c 2 = — — — g44 , or, vice versa, — gu plays the r61e (apart from an additive 2 constant) of the potential multiplied by • c 2 The second terms look less familiar. But their meaning can be made clear at once. They represent at any rate a certain acceleration field which need by no means be negligible in comparison with the newtonian one. The contributions of this field to the components of acceleration are r dg ix dx 2 / dgu _ dg 42 \ _ dx z / dg 43 _ dgq X~\ L dt dt \ dx 2 dxi '■ dt V dxi dxz / -I or in ordinary vector language, with r=(xi, x 2 , x 3 ) and &4=(g41, g42, giz) d*v dgi XT dr = c — cv curl g 4 . dt 2 dt dt Equations <>\ Moi ion This Is manifestly the acceleration due to .1 velocity Held ^gi impressed upon the system of reference. It this velocity field is homogeneous and constanl in time, its contribution to acceleration is, of course, zero; bill if it is heterogeneous and variable, it contributes to the acceleration oi .1 free particle through its time rate of variation auo! through the vorticosity of its distribution. The simplesl case occurs when g., i- ,i linear function of the coordinates alone, say g\\ = -#2i 242= *li J?43 = , C C where w is a constant. Then c curl g 4 is a (three-) vector of size 2o> directed along the x 3 — axis and the last equation gives d-xi _ 9 - dx 2 d-x* 9 . dx } d 2 x 3 _ dl 2 dt ' dt 2 " dt dt 2 which is the Coriolis acceleration corresponding to a uniform rotation of the system with angular velocity o> round the aca - axis (vectorially, with the angular velocity — . curl g 4 ). The reader will, perhaps, miss the centrifugal acceleration C) 2 r, Coriolis' faithful companion. But this (having a scalar potential) is inseparable from g 44 . It is included in g 44 through o>f 2 the term , already familar to us from a previous example. c 2 The ga just given will be found by noticing that in (5), p. 30, rHBdXi— (x\d Xo — x-idx\)dxi. This settles the question. In the more general case the spin \c. curl g 4 will not be constant but will vary from point to point giving rise to a more complicated acceleration field.* The approximate equations of motion (23) can now be written compactly, in three-dimensional vector language, dt 2 2 f d l± - V * curl g, 1 . (23a) L dt dt J *I propose to call so all fields corresponding to any ds-, and to reserve the name of gravitational fields for those only which are 'permanent' or cannot be transformed away holonomously. 38 Relativity and Gravitation" This equation brings at once into evidence the parts played by g 44 and by the three ga condensed in g 4 . Both roles may be equally conspicuous, and it would certainly be unjust to say. with Einstein, that it is only gu which survives in this first approximation. Einstein (loc. cit., p. 817), in deriving the approximate newtonian equations from the rigorous ones, no doubt, through a too hasty computa- tion of the Christoffel symbols, dropped altogether the second terms of our equations (23). And his 'slip' crept into the writings of de Sitter, Weyl and others. Einstein exclaims even (ibid.) in genuine surprise: EdrX{ c 2 dgu ~~1 = — — is that onlv df- 2 dxt _ the component gu of the fundamental tensor determines by itself, in a first approximation, the motion of a material particle'. We shall return to these approximate equations of motion later on, after having set up Einstein's gravitational field- equations. CHAPTER III. Elements of Tensor Algebra and Analysis. 12. In order to be able to construct generally covariant laws or equations, such as Kinstein's field-equations which will complete the fundamental part of his theory, some elementary notions of the Tensor Calculus are required. These I shall now proceed to give, without stopping to sketch the history of the origin and the growth of this powerful method of multidimensional analysis, which the reader will find in the preface to Ricci and Levi-Civita's paper on the Absolute Differential Calculus,* as the said branch of mathe- matics is called by these authors. The relations and properties which are now to occupy our attention hold for a manifold of any number of dimensions. But, if not otherwise stated, we shall have in mind our four- dimensional world or space-time. A world-point is given by four gaussian coordina; which, in general, are mere numbers or labels. As such they need not, as in the most familiar treatment, stand for such things as lengths or distances, or angles. By calling them 'labels' we do not mean, of course, that tetrads of numbers are being haphazardly, disorderly, attached to various events *G. Ricci and T. Levi-Civita, Methodes de calcul differentiel absolu el leurs application, Mathem. Annalen, vol. 54 (1900), pp. 125-201. A con- densed account of this paper is given in J. E. Wright's Invariants of Quad- ratic Differential Forms, Cambridge Tracts, No. 9 (1908). Perhaps the easiest presentation of all that is required for relativistic applications is given in the second part (B.) of Einstein's own paper, loc. ci!., essentially reproduced in chap. Ill of A. S. Eddington's Report, Phys. Soc. London, 1918. Th subject is treated on original and very attractive lines by H. Weyl in Rauin, Zeit, Materie (Springer, Berlin), 3rd ed., 1920. For geo- metrical applications the first volume of L. Bianchi's Lesion; di Geon: Differenzialc (.Spoerri, Pisa), 2 ed., 1902, can be most warmly recommended. 39 40 Relativity and Gravitation (world-points), but we assume that Xi = 7, say, is a label attached to a whole connected three-dimensional continuum of world-points, and similarly for all other (real) numerical values of x%. Likewise for the remaining coordinates, so that every world-point appears as the intersection of, or element common to, some four hypersurfaces of three dimensions. Manifestly, the use of such coordinates does not presuppose any idea of measurement. Again, in this abstract treatment of tensors as certain entities in the manifold, the question whether any one of the coordinates or its differential is space- like or time-like, is of no interest. It becomes relevant only when we come to apply these concepts to physical problems. 13. Such being the nature of the x L , pass from these to any other coordinates x/, through any holonomous transforma- tion whatever, satisfying only the conditions of continuity, etc., as stated in chapter I. Then, as in (8a), the differentials dx„ i.e., the coordinates of a world-point Q, a neighbour of P(x L ), with P as origin, are transformed into 7 . dxj 7 dxj 7 , dx L ' 7 dx l — — - dx K = dx\ + dxi -y . . . . dx K dx\ dx% That is to say, the coordinates of Q with P as origin, are given, in the new system, by these linear homogeneous transforma- tions of the old relative coordinates of the pair of points, with coefficients, dx//dx K , which are some given functions of the position of P. Such an ordered point-pair, PQ, or the corresponding array of the dx L , is called a vector, in our case a four-vector or world-vector. From a more general standpoint to be explained presently its name is: a contravariant tensor of rank one. Now, as in special relativity every tetrad which is trans- formed as the cartesian x, y, z and ct (i.e., by the very special, linear, Lorentz transformation), so here the tetrad of infini- tesimals dx L is made the prototype of all (contravariant) vectors. In other words, every tetrad of magnitudes A 1 which are transformed by the same rule as the dx„ i.e., dx' A' l =°^-A K , (24) dx K Covariani \'i < ro] 41 is called a conlravariant vector or a contravariant tensor of rank one, and A\ A' l i, etc., are called its components. (The upper position of the suffixes was proposed by Ricci and Levi-Civita and accepted by all authors. To be consequent one would have to write also dx 1 , as in fact is done by Weyl. But, for the sake of typographical convenience, an exception is being made for this prototype of all contravariant vectors.) It is scarcely necessary to say that, unlike the Cartesians in special relativity, the coordinates x t themselves do not form a vector; only their differentials do. In short, there are, in general, no finite position-vectors, but only differential ones. This, how- ever, does not exclude the possibility of other finite vectors A 1 . It is of particular importance to notice the linearity and homogeneity of the transformation formula (24) which will reappear in the case of all other tensors. The all-important consequence of this property is that if all components of a vector vanish in one system, they will vanish also in all other systems of coordinates. More briefly, if a vector A K vanishes in one system it will vanish also in any other system. Thus A K = will be a generally 'covariant' or, technically, contra- variant law. This, of course, does not prejudice the question whether Nature is going to obey it. Manifestly, if A" and B" are two contravariant vectors, so also are A K +B K and A K -B K . As dx, served as the standard of contravariant vectors, so do the operators (differentiators) dx, serve as a prototype of another kind of vectors. We have, evidently, dx dx/ and every tetrad of magnitudes B t which are transformed according to this rule, B/=p!LB K1 (25) dx, 42 Relativity and Gravitation is called a covariant vector or tensor of rank one. In com- parison with (24), notice that the suffix of B' coincides with the lower (instead of upper) suffix in the coefficients. Although the prototype of these vectors consists of differentiators, the components B l of a covariant vector need not be operators, but may be magnitudes in the ordinary sense of the word. As in the previous case, B K = is a generally covariant equation or rather set of equations. And if B K and C K be two covariant vectors, so also are B K ±C K . Needless to say that A K -\-B K is neither a covariant nor a contravariant vector. In fact, it has no meaning if the system is not specified. 14. But, while the sum of a covariant and a contravariant vector is from the present point of view of no interest, the combination of their components A l B=A 1 B 1 -\-A^B 2 +A 3 B 3 +A i B i , which is called the inner or scalar product, has a very remark- able property. It is invariant with respect to any transfor- mations of the coordinates. In fact, by (24) and (25), A "B t ' = ^A k ^±B x = (— x ^') A K B X . dx K dxj \dx/ dx K s But the Xi, x 2 , etc., being mutually independent, the bracketed expression (to be summed over all t) vanishes for all k=f X and equals 1 for /c = X. Whence, A n B! = A K B K = A l B„ (26) which was to be proved. Any invariant, S=S r , is also called a scalar or a tensor of rank zero, since, in a manifold of n dimensions, it has n° com- ponents, i.e. but one component. Similarly, a vector or tensor of rank one, has n l = n, in our case four, components. The question whether a scalar is a contravariant or a covariant tensor is idle. For it transforms into itself. Vice versa, it can easily be proved that if B K be four (generally, n) magnitudes such that A K B K is invariant for any contravariant A K , then B K is a covariant vector. And Second k.wk Ti 13 the same thine, is iruc if ' co variant' and 'contravariant* he exchanged with one another. The product of a vector by a scalar is, obviously, again a vector of the same kind, and any number "I v< . tora oi same kind multiplied by scalars and added tog«il again a vector of the same kind. Finally, notice that AJB, and A K B K are not invariant, and thus are no tensors at all. 15. As we just saw, the inner multiplication of a covariant and a contravariant vector degrades the rank of both factor- giving a tensor of rank zero, a single component. Consider, on the other hand, what is known as the outer product of two vectors, of the same or of opposite kinds, i.e., A,B K , or A l B", or A t B K . The suffixes being here different, no summation i> understood, so that each of these symbols stands for 4 2 =1G (generally w 2 ) components. Let us take A L B K first, which is a short symbol for the array AxB x ArB 2 . . . A 2 B y A2B2 . . . of sixteen magnitudes. Denote them by M^ respectively. Their law of transformation is, by (25), M' m -p-pLM+ (27) dx L dx K Every array of n 2 magnitudes N lK (whether obtained by the outer multiplication of two covariant vectors or in any other way) which is transformed by the rule (27) is called a covariant tensor of rank two. It manifestly has again the property of vanishing in all systems, if it vanishes in one of them. In a four-manifold N lK consists of 16 components. In general N lK + N Kl . If, in particular, N lK = N Kl the tensor is called symmetrical. An example of such a tensor we had in g lK , called the fundamental tensor; cf. formula (9). Notice, however, that the tensor property of g lK followed from the invariance of ds' : which fixed the metrical properties of the world, whereas all our present considerations are entirely independent of the —4 44 Relativity and Gravitation metrics of the manifold, and it is preferable to abstain from using them at this stage. Such properties as are impressed upon the general tensors by the metrics of the world will be treated in later sections. In the meantime let us continue the non-metrical theory of tensors. The symmetrical tensor N lK consists in general of |w(w+l), and for w = 4, of ten different components. It can be easily proved, by (27), that its symmetry is an invariant property, i.e., that if N lK =N Kl in one system, we have also N f lK =N' Kl in any other system. A covariant symmetrical tensor of rank two can be constructed at once from a covariant vector, to wit by forming its outer self-product, A llv = A ll A v = A v A lli = A Vfi . If N lK = —N KL , for all i, k, we have an antisymmetrical (or skew) tensor. Since N KK =—N KK means N KK = 0, a whole diagonal of components vanish, and thus only ^w(w-fT) — n = %n(n—l) non-vanishing and independent components are left, the surviving ones being oppositely equal in pairs. Thus an antisymmetric tensor in a four- world consists of six independent components, and is therefore called a six- vector, in the present case a covariant six-vector. With such six-vectors the reader is already acquainted from the special relativistic treatment of the electromagnetic field. We shall see them at work in a similar duty in general relativity later on. As the symmetry so also the antisymmetry is an invariant property, i.e., iV tK = — N Kt is transformed into N' tK = — N' KL . Any tensor N tK can be split at once into a symmetrical and an antisymmetrical one. For we have identically N lK =i(N lK +N Kl )+UN tK -N Kl ), and the first term represents a symmetrical, the second an antisymmetrical tensor. Similarly to (27), and starting from the special tensor A l B K , any array of n 2 magnitudes which are transformed by the rule N ,iK = -^ bx -L. N aP (27a) dx a dxp Mixed Tensors 15 is called a conlravarianl tensor of rank two. If N" = A*", it is a symmetrical, and if N"= —A 7 " 1 , an antisymmetrical tei (A tensor A 7 "' need not be the product of two contravarianl vectors.) Lastly (starting from AJ¥), any array of n- magnitudes A 7 ," which are transformed by the mixed rule N , K = dx^ dxj_ N * (276) dx/ dx a is called a mixed tensor of rank two, covariant with respect to its lower suffix t, and contravariant with respect to its upper suffix or index k* Special cases of symmetry and anti- symmetry as before. A new feature, however, offered by the mixed tensor is this. With any A 7 " make t = /c, getting N K K and, by the usual convention, sum over all k. In other words add up all the components of the chief diagonal (slanting down from left to right) of the mixed tensor. The result will be a single magnitude. Now, the important thing is that this magnitude is a general invariant. In fact, by (27b), \dx \. dx n / but (as mentioned before) the bracketed expression is zero for all a?±fi and one for a = fi. Thus N'l=Nl=Nl, which proves the proposition. Thus, equalling the upper and the lower index and summing over it degrades the mixed tensor by two rank- giving, in the present case, a tensor of rank zero or an invariant (scalar). In other words, n:=n *It seems inappropriate to call 'surrix' (from sub, under) an upper mark or sign. I propose .therefore, to call such signs by the more general name ■index. Since all English writing authors accepted the ' three-;;; dex symbols ' and the 'four-index symbols' (of Christoffel and Riemann), they will per- haps not object to calling t, k indices. 46 Relativity and Gravitation is an invariant of the tensor N K ,. We shall see presently that this procedure of equalling an upper to a lower index, called contraction (German ' Verjiingung') can be applied, with equal success, to a mixed tensor of any rank whatever. Notice, however, that this process is not applicable in the case of (purely) covariant or contravariant tensors. Thus, for instance, M KK = M n + M22 + ... is not invariant, as a glance on (27) will suffice to show. In short, the diagonal sum of M lK has no intrinsic meaning. Similarly, in the case of a four- vector, say, Ai~\- . . . -\-A± is not an invariant. 16. The next step, leading to tensors of rank three, and so on, is obvious. Generally, any system of w r (in our world, 4 r ) magnitudes iV*. ';;, with r\ lower and rz upper indices, which are transformed by the rule /■pja0...\r _ vXj dXfc OX a OXp /y? 6 --- (28") dxj dx K ' dx a dx b is called a mixed tensor of rank r = ri-\-r 2 , covariant with respect to its r± lower, and contravariant with respect to its r 2 upper indices. If all the components of such a tensor vanish in one system they will also vanish in any other system of coordinates. Any tensor, therefore, can be used for writing down generally covariant laws.* In particular, if ri = 0, the tensor (28) is con- travariant, of rank r 2 ; and if rz = 0, covariant of rank n. The sum of any number of tensors of the same rank and kind, each multiplied by any scalar, is again a tensor of the same rank and kind, the numbers n, r 2 retaining their significance. 17. Contraction. This process, already illustrated on the simplest example, can now be generally explained. Let a be any upper and t, any lower indexf of a mixed tensor of any rank r whatever. Put a = t and sum over a. Then the result will be a tensor of rank r — 2, with r\ — \ covari- ant and 7"2— 1 contravariant indices. *In the less technical sense of the word. •jThe place of a among the upper, and of t among the lower indices is irrelevant. Contraction of Tensob I. The proof follows at once from (28). For the process gives us in the coefficients of transformation a term dx a dxj dx,,' dXi which vanishes for all a^i and equals one for a = i, thus reducing (28) to (flf;:;)'- **L ^L ....Afc dx/ d.r*. Ar... . which proves the statement. This process of contraction can obviously be applied again and again, degrading the tensor each time by two ranks until there will be no upper or no lower indices left. In fine, tin- mixed tensor can be degraded until it becomes purely covariant or purely contravariant or (if ri = r 2 ) until it is reduced to a scalar or invariant. Thus, for example, the tensor A*j* x of rank five gives rise to A aK \ = ^n\ + A^ K \ + • • • l which is denoted by A% x , and this tensor of rank three gives rise to 4* = ^x which is a (covariant) tensor of rank one or a vector. Again (as an example of r\ = r 2 ) . the tensor A*j* of rank four gives by contraction A% , and this tensor of rank two gives a: = a, a scalar. We may as well write at once A™ = A, the meaning and the value of A being the same as before. This final invariant may be considered as a property of the original tensor A"? . In general every such half-and-half tensor (fi=rs) will have the final scalar (A) as its intrinsic* invariant. And, as far as I can see, this is its only intrinsic invariant. *i.e. an invariant of its own, independent of any extraneous form such as ds 2 (or any auxiliary tensor, such as g lK ) determining the metrics of the manifold. 48 Relativity and Gravitation On the other hand a purely covariant or contravariant tensor or an unequally mixed one (fi?^) cannot be contracted to an invariant. It seems that it has no intrinsic invariant at all, that is to say, that there are no processes which would lead to an invariant combination of the components of the original tensor itself (without using other tensors). 18. The inner multiplication, already mentioned in con- nection with vectors, can now be considered as an outer multiplication followed by a contraction. Consider two tensors, generally mixed, one of rank r = ri~\- r 2 , the other of rank 5 = 5i+52- Combine (by ordinary multi- plication) each of the n r components of the former with each of the n 3 components of the latter. The n r+s magnitudes thus obtained will be the components of a tensor of rank r+s with fi+Si covariant and r 2 +s 2 contravariant indices. That the entity thus arising is a tensor follows at once from (28). Thus the outer product of two vectors is a tensor of rank two, A t B K = M tK , A L B K = M K L . Similarly A afi B lK is a covariant tensor of rank four, M a p lK , and A aP By = N L ^ y is a mixed tensor of rank five, and so on. The outer multiplication combined with contraction (when there are indices to contract) gives the inner product. Thus the inner product of A L and B K is A K B K =M K K = M, an invariant.* The inner product of A K and B a $ is their outer product M K a p degraded by contraction, i.e., M% = M a , a covari- ant vector. The inner product of A a p and B lK is their outer product A a pB lK = M l *p degraded (to the utmost) by two contractions, m k :=m, i.e., a scalar or invariant. Vice versa, if A a p be any array of n 2 magnitudes such that A aP B lK is an invariant for any con- travariant B lK , then A a $ is a covariant tensor of rank two. This criterion of tensor character, already mentioned in con- nection with A L B K , can be easily proved by writing down the There is no inner product of A L , B K . Tensor Differentiation 19 transformation formula of the given factor (tensor). And it can be extended to any rank and kind, no matter whether the inner product is a scalar or a tensor of any rank higher than zero. As we already know, the differential operators D t = • have the character of the components of a covariant tensor of rank one. Therefore, the 'product' of this tensor into a scalar or scalar-field f=f(xi, x 2 ■ ■ ■), that is to say, the result of operating with Dl upon/, will again be a covariant tensor of rank one or a covariant vector, ^- = A u . (29) Ox, But we cannot go further than that. That is to say, an iterated application of the operation D K does not give a tensor. Thus d' 2 f/dx L dx K is not a tensor. Nor do, in the more general case of any vector B u the n 2 derivatives D K B t = dB L fdx t constitu te a tensor. The different behaviour of D K B l and of products of magnitude-tensors lies herein that the operational tensor D K acts also on the coefficients dx K /dx,_' of the transformation formula of B t . In fact, we have DW-*±D m (23LBd, dx K ' V dx t ' / and* this is not the same thing as —2 —?- D.B 8 . The same dxj d\\' remark applies, a fortiori, to higher derivatives of scalars and of tensors of any rank. In fine, the only tensor derivable by simple differentiation, unaided by other auxiliaries (cf. infra), is the covariant vector (29) yielded by a scalar. The vector or vector-field df d.\\ is called the gradient of/. In the case of space-time it consists of four components. *Unless the coordinate transformations are linear as in the spoci.il relativity theory. 50 Relativity and Gravitation 19. Tensor properties in a metrical manifold. Having sufficiently acquainted ourselves with the properties of tensors in themselves, let us now consider them in relation to the fundamental quadratic form ds t = g lK dx,dx K which converts the hitherto amorphous world into a metrical or riemannian* manifold. It is of the utmost importance to grasp well this distinction between a riemannian and a non-metrical manifold and to understand the true role of ds 2 in converting the latter into the former. Let us place ourselves yet for a while upon the non-metrical standpoint. Of all the tensors described in the preceding sections let us confine our attention upon the prototype of all (contravariant) vectors, the infinitesimal position-vector dx t . Any such vector represents ultimately but an ordered pair of points, 0(x L ) the origin, and A(x l ~\-dx l ) the end-point of the vector. Imagine a whole bundle of such infinitesimal vectors OA, OB, OC, etc., all emerging from the same world-point O as origin. Now, from the non-metrical point of view, all these vectors have (apart from their origin) nothing in common with one another. That is to say, if two of them, say OA and OB, are at all distinct from one another, and if their components dx L do not happen to be proportional to one another (in which case we can say that the vectors have a common 'direction'), there is in either of them nothing, no property, with respect to which they could be compared. They are, as it were, perfect strangers to one another. Similarly, if we call 'angle' a vector-pair a — OA, OB, there is nothing to base upon a comparison of two non-overlapping covertical angles a and (3 = OC, OD. In short, neither vectors nor angles (or other derived entities) have 'sizes'. There is, in fact, in the manifold itself nothing which could fix the mere meaning of such a concept. Of two vectors OA, OB nothing more can *The name 'riemannian' manifold or w-space is being often used in this connection in view of the historical fact that Riemann was the first to base the general geometry of an rc-space upon its line-element given by such a differential form, although Gauss was his great predecessor in the case of surface theory. The Line-Elemi ~>\ be said than thai they arc either identical (or co-directional, collinear) with or distincl from one another. The origin being the same, 11 the points A , B are either identical or dis- tinct, and no other significant statement ran be made ab their relation. But while there is nothing in the manifold itself to base a comparison of distinct infinitesimal vectors upon, we an liberty to provide for it at our will if we so desire. This ifl done by introducing a standard or fundamental entity such as the quadratic form called the line-element. In other words, we surround the world-point 0{x t ) by a hypersurface, a three-dimensional (generally an n — 1 dimensional) quadric and declare all vectors emerging from and ending in any point P {x L -\-dx) of this surface to be equal in size or in absolute value, or in 'length', the usual name in the case of our three-space. It is precisely this metrical surfacef which is expressed by g lK dx t dx K = ds 2 = const., the numerical value of ds being the 'size' common to all these infinitesimal vectors or point-pairs. J The part played by this quadratic form is essentially the same as that of Cayley's 'absolute' or standard quadric (a real quadric leading to lobatchevskyan or hyperbolic, an imaginary quadric leading to elliptic, and the intermediate degenerate quadric leading to euclidean geometry), the only important difference being that Riemann's treatment is much more general. It covers *We have limited the discussion to coinitial vectors solely for the sake of simplicity. All our remarks apply a fortiori to distant, non-coinitul bundles of vectors. tThe German geometers call it Eichflciche. Jin Riemann's own treatment this role of the fundamental form im- pressed upon the manifold extends into distance, over all the manifold. That is to say, if 0'(y ) be any other point and if a quadric g _dy dy = const, be drawn around it with the same value of the constant as before, all the vectors of the bundle O' terminating upon this quadric are again said to have the same size as those of the bundle 0. In this respect a somewhat more general standpoint was recently proposed by Weyl, in connection with his ideas on electromagnctism. 52 Relativity and Gravitation all metrical spaces (in technical language, of variable and anisotropic curvature), whereas Cayley's device gives us only a space of constant isotropic curvature, negative, zero, or positive. This fully corresponds to his starting point, which was that of projective geometry. Yet, and this is of particular interest in the present connection, Cayley recog- nized thoroughly the true role of all such standard entities. In fact, he tells us plainly that geometrical figures have no metrical properties in themselves. Their metrical properties such as those of the foci of a conic, etc., arise only by relating them to other figures, as the 'absolute' conic in the plane, or quadric in three-space. The kind of metrics thus impressed upon a continuous manifold being essentially arbitrary, the utility of the metrical manifold thus obtained will, of course, from the physicist's standpoint, depend upon the interpretation which is given to the said 'size' of a position-vector, and to special lines of that metrical manifold, such as the geodesies, in terms of measuring rods, clocks, moving particles or light phenomena, and so on. But without dwelling here any further upon such questions of a concrete representation let us turn to consider the purely mathematical consequences of the introduction of g tK dx l dx K as a fundamental differential form fixing the metrics of the manifold. 20. As in Cayley's case the geometrical figures in relation to his 'absolute', so here the tensors acquire some new pro- perties in relation to the fundamental form or better, to its coefficients g lK . In fact, what determines the form are these coefficients, and we may look upon the matter in the following way. Instead of declaring the fundamental quadratic form at the outset as an invariant, let us better say that the symmetrical array of 16 (generally ri 2 ) magnitudes g lK is being introduced as a. fundamental tensor, symmetrical, of rank two and of the covariant kind, as defined in the preceding sections. Combined with this fundamental tensor all other tensors of the previously amorphous manifold will acquire some Metrical Properth mcv. properties. These and only these will now l»<- their metrical properties. To begin with the prototype of contravarianl ve< tors, the infinitesimal vector dx t has had thus far no invariant of his own. Hut it will acquire one with the aid of the fundamental tensor. In fact, dx l being contravariant, denote it for the moment by X 1 . Form the outer product which will be a mixed tensor A '^ . Contract it with respect to i, a, getting A\p=Ap. Contract this again. Then the result will be A K K = A, a scalar or invariant. Or perform both contractions at once, and write ds 2 for A, returning to the original notation, thus g lK dx L dx K zEzds-= invariant. In short, the inner product of the tensor dx a dx$ into the funda- mental tensor g lK is an invariant. There is no objection to calling it the invariant oj dx t as a short name for its metrical or associated invariant. Thus, thanks to g u , the vector dx, has acquired an invariant. And it can now be compared through it with other vectors, no matter what their com- ponents. The value of ds 2 may be called the norm, and the absolute value of v ±ds 2 the size of the vector dx\ . Thus we can speak of two vectors dx t and dy t being equal in size, or one having twice the size of the other, and so on. In application to the four- world, a vector dx t of no size will be a light vector, a vector of negative norm a space-like, and one of positive norm a time-like vector. Similarly, any other contravariant vector A' will have the metrical invariant gut A'A K =A\ say.* (30) *Of course, even in the amorphous manifold an invariant could be built up from A 1 by the aid of any covariant tensor N lK , but the choice of W M being entirely free, such an invariant would not have a fixed value. We fix it by introducing once for all a special tensor g lK to serve for all other tensors. 54 Relativity axd Gravitation In much the same way. if B L be any covariant vector, we shall have in g^B.B^B- (30a) an invariant, the norm of B L . From a more general point of view we may call -4-. in (30). the tensor, of rank zero, metrically associated to .4', similarly, in (30a), B- to B L . Moreover, we can easily construct associated tensors of a rank other than zero, and differing also in kind from the original tensor. Thus, to dwell still upon vectors, g^A' = A t (31) will be the covariant vector metrically associated with the contra variant vector A'. We may call A, shortly the conjugate of A\ Similarly, starting from a covariant vector A tt we shall have the contravariant vector g a A K = A' (31a) conjugate to -4, . Two questions naturally suggest themselves: Will the con- jugate of the conjugate be the original vector? Have two conjugate vectors the same size or the same norm? In order to answer these questions as well as for the sake of what will follow, let us first note a simple property of the tensors g^ and g 1 * . By definition, chap. II. g" is the minor of the determinant g = g„ . corresponding to its i, K-th element, divided by g itself. But g is equal to the sum of the produce of the elements of its first column, say, into the corresponding minors, i.e., g = g a igg ai . whence gaig° 1= l- Similarly for any other column (or row). Thus, underlining the index over which an expression is not to be summed. This is valid for even.- v. Thus g^g"", summed over both indices, has the value 4 for our world, and n for an ?i-fold. Again, taking two different columns (or rows) of g. we shall easilv prove that ( onji GATE 'I i NSOBS 56 Both properties can be united in formula «..«•' = *! = «!. where 6„ is the conventional symbol for 1 or according as a = /3 or aj^/3. This symbol is itself a mixed tensor. We are now able to answer our two questions. First, the conjugate of the conjugate of the vector A t is, by the definitions (31), (31a), n g" A = g l A = 6' - 1 = 1 i.e., the original vector. Similarly if we started with A*. Thus, the conjugate of the conjugate is the original vector. Second, if A 1 be the conjugate of A t we have for the norm of the former vector, by (30) and (31a). g^A'A^g^g" A mj f'A fi =8 a K A^'Ap = £'A fi A m . Thus any two conjugate vectors have equal norms. The norm of A t and of A' can also be written A t A l , for this is again equal to g^ A l A K . Thus, for instance, if d^ be the conjugate of the contravariant vector dx u their common norm or the squared line-element can be written ds- = d.\\d^. (33) 21. In much the same way we can treat the metrical properties of tensors of any higher rank. To explain the method it will be enough to take up in some detail the second rank tensor A u . Its conjugate or supplement (Erganzung) will be the contravariant tensor defined by ffA^-A' , or also g„a«4*-it« . (34) The tensor g** itself is easily proved to be the supplement of the tensor g lK . The scalar or invariant of A^ will be CA m -Al-A. (35) A single contraction of g" A a& will give o" A =.4 l a mixed tensor metrically associated with the covariant A~. 56 Relativity and Gravitation The supplement of the supplement (or the conjugate of the conjugate) is again the original tensor, for g ai getA" = g ac g 0K g> 1 g SK A yi = dl 8 5 eA y& =A yb The tensors A lK and A" have the same scalar A, (35). In fact, the scalar of A lK is U A"" =&, C 2^.* = E t A aP = i K Az=A. Since g? v A MV is an invariant, B^ =g lK g?" A liV is again a tensor; Einstein calls it the reduced tensor belonging to A^,. Notice that neither a covariant nor a contra variant tensor has an invariant independent of the metrical tensor; only a mixed tensor, B K has such an invariant, to wit B = B K . This is a privilege of mixed tensors of even rank with ri=r 2 , and of these tensors only. The investigation of other metrical properties of tensors of the second and higher ranks may be left to the reader. Exercises of such a kind will soon make him familiar with this broad and powerful algorithm. 22. Angle and volume. Consider an) 7 two coinitial in- finitesimal vectors dx L , dy L . These are contra variant vectors. Therefore, as we already know, the inner product o dx dv Sue """*i "'.»« will be an invariant. It will remain invariant when divided by the sizes of both vectors. By an obvious generalisation of the familiar cosine formula this invariant is used to define the angle e made by the two vectors, thus g IK dx t dy K . . cos e = , (ooj ds do- where ds 2 = g LK dx t dx K , da 2 = g LK dy t dy K , The two vectors are said to be orthogonal or perpendicular upon one another if g LK dx t dy K = . Generally, the angle between any two vectors A\ B\ whose norms as defined by (30) are A 2 and B 2 , will be determined by i.k and Volume cos t = g« A % BT AB and the vectors will be orthogonal if g„ i4\4*=0. Similarly for covariant vectors, with the only difference th.a ■■ replaced by g lK . Let A t , B K be the conjugates of A*, B l ; then g 1 " ^B.-g" &. g^A a B" = 5l g fiK A a B fi = g afi A a B", and since -<4 t , B l have the same norms as A 1 , B\ we see that the angle between the conjugates is the same as between the original vectors. The integral fdx\dx% ■ ■ • dx n extended over a domain of the manifold is, by a well-known theorem, transformed into fJdxi dx% . ■ ■ dx n \ where / is the Jacobian a.v/ , as in (7). On the other hand the determinant g of the fundamental tensor (called also the discriminant of the fundamental quadratic form) is transformed into dx a dx fi dxj dXp ~dx7 the last step being based on the multiplication rule of deter- minants. Thus a' = J*o, Consequently, the integral fVg dxidxi dx, (38) (39) is a scalar or an invariant of the n— dimensional domain of integration. In the case of the four-dimensional world the determinant g is always negative.* Thus the invariant expression *In a galilean domain and in Cartesians g= — 1, by (lb), p. 6. Bj therefore, it is also negative, always for a galilean domain, in any other system of coordinates derived from the Cartesians by a holonomous t: formation. Now, although a non-galilean domain cannot be made galilean by a holonomous transformation, yet we know that in all practical C the g lK differ but very little from the galilean coefficients. Thus f will also in general be negative. 58 Relativity and Gravitation dQ, = v — g dx\ dxt d%z dxi (40) will be real. This is taken as 'the local measure' of the size or volume of an infinitesimal world-domain. For in the local (cartesian) coordinates u lt for which g= — 1, this expression becomes du\du^duzdu^ = cdtdxdydz. The latter product is called by Einstein 'the natural' volume-element. Apart from names, the important thing to notice is the general invariance of the expression (40) as such or when integrated over any world-domain. Consider any sub-domain of the world, of three, two or one dimension. This can be represented by expressing the x t as functions of three, two or one parameter respectively. The differentials dx L will be homogeneous linear functions of the differentials dp a of these independent parameters. Thus the line-element within the sub-domain will be of the form ds 2 = /fo0 dp a dpp , h a p = h@ a , and the sub-domain, therefore, will again be a metrical manifold (a three- space, surface or line) in Riemann's sense of the word, and if /z = |Zt a /sl» d£L=V h dpidpz . . . will (apart perhaps from a factor V — 1) again be an invariant measure of an element (volume, area, length) of the sub-domain. Thus, in the case of a one-dimensional sub-domain or line, j dx L dx^ = dp, dp and ds 2 =g lK r-—^- — — dp 2 = hu dp 2 , say. dp dp In this case h = hn and, therefore, dtt=V~h u dp, which is ds itself, as it should be. For a two-dimensional sub-domain or surface we have ds 2 = hn dpi 2 +2hu dpi dp2+hi2 dpi 2 , dx, dx K where h a b=guc ~T— "TT - • opa upb Thus, where —4 dtt=V h dpidp 2 , 3*i dx K dx L d-\ _ / dx t dx K y dpi dpi tK dp2 dpi \ lK dpi dp2 / < 0VARIAN1 I >him \ i|\ I 23. Differentiation based on metrics. We have already (p. 49) thai if/ be a scalar or invariant, df/dx„ the gradienl of /', is a covariant vector. This is independent of the metric - of the manifold. But, as was then pointed out, the iterated application of the operation d t dx, would not lead to nor would its application to a vector A, or another tensor yield by itself, unaided by auxiliaries such as g u , a tensor. But the introduction of the metrical tensor opens in this respect new and important possibilities. It was remarked by Christoffel as long ago as INfi'i thai if A t be a covariant tensor, so is J r:K ox k namely covariant, of rank two. Similarly if B lK be a covariant tensor of rank two, ** _ "5T ~ l a/ B "-\ a i B - ' 42 ' is again a covariant tensor of rank three; similarly dx x I « J ( K I ^ = _^_ _ .a b;+ ^ . ^ (42fl) is a mixed tensor of rank three, and so on. But it will be enough to consider here at some length the first case (41) only, especially as the other cases can be derived from it. The operation indicated in (41) is called covariant differentia- tion, and its result A M the covariant derivative or the expansion (Rrweiterung) of A L . 1 1 B' be a contra variant vector, B" =e is a contra variant tensor of rank two, the cotitravariant derivative of B. But for our purposes it will suffice to consider only the covariant differentiation. That (41) represents a covariant tensor can be proved in a variety of ways. The most instructive of these is perhaps 60 Relativity and Gravitation that given by Einstein, since it makes immediate use of the equations of geodesies, and the role of the Christoffel symbols* appearing in (41) is thus far known to us only in connection with these world-lines. Einstein's reasoning is as follows: Let/ be a scalar or better a scalar field (i.e. an invariant function of position within the world). Differentiate it twice along any world-line. Then d 2 f df d 2 x t d 2 / dx a dxp ds 2 dx t ds 2 dx t dxp ds ds will again be an invariant. Let the line be a geodesic. Then x L j a/3 ( dx„ dxg , , . ... — = — < > — — , and the invariant will assume s 2 { <■ ) ds ds ds 2 the form d 2 f .I d *f / a/3 I 3/ "I dx a dx, = r d2 f - \ a ^\^c\ L dx a dXa I L J dx L -J ds 2 I— dx a dxp \ L j dx L — 1 ds ds Since the contravariant tensor (of rank two) dx a dx$ is arbitrary (for from a given point a geodesic can be drawn in any direc- tion, i.e. with arbitrary ratios of dxi, dx 2 , etc.) and its product into the bracketed term is invariant, the latter, i.e. f =^L.-f«\ll (43) dx L dx K I a ) dx a is a covariant tensor of rank two. This proves the proposition for the special vector A L =df/dx t . To prove it for any covariant vector, notice that any such vector A t can be repre- sented by the sum of four (generally n) terms of the form \pdf/dx t , where \p and / are scalars. Thus it is enough to prove that dx„ \ dx t / \ a ) dx„ is a tensor. But this is equal to *Notice in passing that ^ > is not a tensor. X j (ovarian i Differentiation 61 «. + ■*-*- OX, ()X K which, being the sum of covariant tensors of rank two itself a tensor of the same kind and rank. Thus the tensor character of the derivative (41) of any vector A, is proved. Notice that for constant g„ (galilean world) the Christoffel symbols vanish and this covariant tensor of derivatives reduces to an array of ordinary^dem a- tives dAJdx K . The proof of the tensor character of (42), which can be easily deduced from that of (41), may be left to the care of the reader. It is interesting to note that the covariant derivative of the metrical tensor g lK itself vanishes dentically. In fact, substituting in (42) g LK for B lK we have and since i< which will also be useful in other connections, and similarly for the last term, we have 3g« ["]-[*. X ] dxx But by the definition (13) of the symbols, and since g = g_ , we have Thus, g LK \ = 0, identically. Let A lK be as in (41), where A, stands for any covariant vector. Then, since the second term in (41) is symmetrical in i, k, dAJdx K — dA K /dx L = A lK —A Ki , being the difference of two tensors, is again a tensor. This tensor is called the rotation of the covariant vector A t , and can be written Rot (.4.) = i^L _ Mi. (45) dxp dx a 62 Relativity and Gravitation This covariant tensor of rank two is manifestly antisym- metrical, i.e., in the case of a four- manifold, a six-vector. Notice that although the proof of the tensor character of the rotation was based on the metrical formula (41), yet the rotation itself, as defined by (45), is entirely independent of the metrical properties impressed upon the manifold. It contains no trace of the metrical tensor g LK . The same is true of a tensor of rank three which can be deduced from (42). Let in that formula B lK be an anti- symmetric tensor or six-vector. Then B^+B Al +B^- ***- + «3si + 23* . (46) d%x ox t dx K Thus the right hand member is again a tensor. This is called the antisymmetric expansion of the six-vector B tK . It will, together with the rotation (45), be of use in connection with electromagnetism. Another tensor derived from a six-vector of equal import- ance in the said connection is ^ = VTg.£" (V-~g^) = DivG4' K ), (47) a contra variant vector, called the divergence of the contravariant six-vector A tK = —A" . The proof of its tensor character, to be based on (42), can be omitted here. Finally let us mention, without proof, that ^7 ^7 (v:ig^)=div(^), (48) called the divergence of the contravariant vector A K , is a scalar or invariant. 24. The Riemann-Christoffel tensor is of such capital im- portance for Einstein's gravitation theory, and for the geometry of any riemannian manifold, as to deserve to be treated at some length. It is a metrical tensor of rank four, built up of g tK and their first and second derivatives, known to the general geometers since the time of Riemann. The ( Curvature Tensob I i expresses the so-called curvature properties oi .1 mani- fold or n-space whose metrical relations arc fixed by th< &, , and to Einstein il served as the material for building up his gravitational field-equations. In order to arrive at this all-important tensor let US .>tart from an arbitrary covariant vector A t and let us write down its second covariant derivative, that is to say the covariant derivative of the tensor A lK which is the covariant derivative of A„ i.e., by (42), _ dA„ f i\\ ( k\\ , dx x ' a ' I a ' where A iK is as in (41). Similarly, transposing k and X, let us write the second covariant derivative I _ dA * _/ tK I a _ / x *l i dx K I. a ) { a ) Either being a third-rank tensor, so will be their difference A a _ dA lK dA lX ) t\ \ ,/**>•) 4 d.\\ <1.\\ { a ) { a ) This is, by (41), LteA aT-J^Xa /J^ + Ll a ft »f~ In. the second term the indices a and /? over which the sum is to be taken can be interchanged. Thus A^ — A^ is the inner product, of an arbitrary covariant vector A a into the sum of the tw r o bracketed expressions. This sum, therefore, (49) ». - & ( t\\ d j ck\ i c\\f ISk\ (ikH0X\ '"" dZ\ a / " ^\ a f + \ fi ! I « J " I (3 A - V is a mixed tensor of rank four. This is the Rieman n-Christoffel tensor which, for reasons to appear presently, may as well be called the curvature tensor. 64 Relativity and Gravitation Strictly speaking, Riemann's own system of four-index symbols (iju> Xk), discovered in 1861 in connection with a problem in heat conduction (Mathematische Werke, 2nd ed., p. 391), is the purely covariant tensor associated with (49), to wit ■^Xk = (tM. Xk) = g M a £?«x ■ ( 50 ) From this we have conversely, K-x = <T(tM, Xk). (50a) Also the latter tensor was used in geometry for a long time.* The Riemann symbols are, for an w-space, w 4 in number, and for our world, therefore, as many as 256. But they are bound to one another by the linear relations (ifJL, K\) = — (/JLL, KX), (ifl, KX) = — (l/JL, Xk), (t/Lt, «X) = (kX, i/x) , (l/JL, /cX) + (lX, /Ik) + ((.K, Xju)=0, so that the number of essentially different, i.e. linearly independent symbols is reduced to N= "W-V . (51) 12 For a proof see, for instance, Killing, loc. cit., p. 228. In the case of a one-dimensional manifold, a line, there is no such non-vanishing symbol. In fact, although a line may be ' curved ' from the standpoint of two- or more-dimensional beings in whose space it is imbedded, yet it has no intrinsic properties of its own to distinguish it from other lines, nor one of its parts from another. Take, for instance, a plane curve. If Aco be the angle between the tangents at two points separated by the arc As, the curvature of the line is defined as the limit du/ds. Now, this curvature is often called an intrinsic property of the line, because (unlike the sloping of the line) it is independent of a coordinate system laid in *Cf. for instance L. Bianchi, 1902, loc. cit., p. 72, where it is denoted by |ia, X/c|. The geometrical applications of the Riemann symbols are fully- treated in vol. I of Bianchi's work. See also W. Killing's Nicht-Euklidiscjie Raumformen, Leipzig (Teubner), 1885. Riemann Symbols that plane, yet it is entirely meaningless it the line is not conceived as a sub-domain of the plane. For bo is the angle - And from the bidimensional standpoint every curve is developable upon every other. In the case of a surface, n = 2, there is, by (51 1, essentially just one Riemann symbol, namely (12, 12), (21, 21) being equal, and (12, 21), (21, 12) oppositely equal to it, and all others being zero. This unique symbol divided by the discriminant g is a differential invariant of the surface (or of its metrical form ds' 2 = g n dxi ? +2gxndxidxo-\- g^dx* 1 ) . This invariant, K = < 12 ' 12 > = n2 - 12 > . g gll g22 - gl2" is the familiar gaussian curvature of the surface, its reciprocal being the product of the two principal radii of curvature. This is an intrinsic metrical property of the surface, requiring for its general definition or its numerical evaluation no refer- ence whatever to a third dimension. In fact, (52) contains only the metrical tensor components g LK and their first and second derivatives with respect to any gaussian coordinate system spread over the surface itself as a network of lines. The curvature thus defined, in general variable from point to point, can be evaluated at any spot by dividing the exc<.-~ of the angle sum (over a straight angle) of an infinitesimal triangle by the area of the triangle. Again, as is well known, the necessary and sufficient condition for a surface to be developable upon a euclidean plane or for its fundamental form to be holonomously transformable into ds 1 = dx- + dy- is the vanishing of K, i.e. of (12, 12)', and herewith the vanish- ing of the whole tensor of Riemann symbols. For a three-space there are, by (51), six, and for the world or space-time as many as twenty independent Riemann symbols. A five-space has fifty independent symbols, and so on. But, no matter what the number of dimensions, the 66 Relativity and Gravitation Riemann symbols always represent the curvature relations of the manifold, and their vanishing" continues to form the con- dition of an important property of the metrical form of the manifold. To begin with the latter,. suppose all g u: are constant over a domain of the world. Then all (in, Xk), and therefore also all the components of the tensor jB". x vanish throughout the domain. This then is the necessary condition for a domain of the world to be galilean, i.e., for the line-element to be holonomously transformable into ds 2 = c 2 df — dx 2 — dy' 2 — dz 2 '. It was proved by Lipschitz that this, i.e. is also the sufficient condition for the said reducibility (to a form with constant coefficients). In the second place, concerning the curvature relations, consider a surface <r or a two-dimensional sub-domain of the world, or of any metrical manifold. More especially, let a be a geodesic surface. This can be defined as follows. From a point 0(x L ) draw two infinitesimal vectors d£ t , drj, and con- sider the pencil of infinitesimal vectors dx t = adi; L -{-(3dri L , where a, (3 are free coefficients. Draw from the geodesic lines with each of these vectors as initial direction. The surface thus constructed will be a geodesic stirface through 0. Its normal v will be defined by the infinitesimal vector dy L orthogonal to d^ and d-q l} i.e., such that guc dy L dL = gu; dy\d VK = 0, and therefore also g lK dy t dx K — 0. The geodesic surface <j = o v will thus be completely determined by 0, one of its points, and by its orientation, given by the normal just explained. The line-element of the manifold (world) will give for the line-element of this surface, at 0, an expression of the form ds 2 = /zn dp 1 2J c2hi» dpi dp 2 -{-Jh2 dpi 2 , and the gaussian curvature of a„ will be, as before, with h = h n h 2 i — /?i2 2 , RlEMANNIAN ( i i'\.\ l I RE 67 K = (12, 12) ; , h >2 This is, according i<> Riemann's definition, the curvature <>] the manifold, at O, corresponding to the orientation v of the surface. The suffix //. has to remind us thai the Riemann symbol is to be formed with h m as the- fundamental tensor (of the sub-manifold a,). It remains to express (52a) in term- of the original tensor g u of the manifold and the vectors ei£,, dr] t defining the orientation of the surface. This gives ultimately* A' = 1 ^' U\, Kn) t where ,SlK fc = 2' .sX/i #. dij t d£ dy K d^ dr)\ d ^ dv,* #. drj t < dr, K dh dv\ #„ d V)l the dashed sums to be extended only over such combination- of the indices for which i<\ and at the same time k<h- This will suffice to show the role of the four-index symbols in determining the riemannian curvature of a metrical manifold of any number of dimensions. In general, the curvature will not only vary from point to point but will also be different for different surface orientations (v). In short, the manifold will in general be heterogeneous and anisotropic with regard to its curvature. Such, for instance, will be the world as the seat of a permanent gravitational held. On the other hand, a galilean domain, for which all (tX, ku) vanish, will have everywhere and for every orientation the curvature zero. In other words, it will be flat or homaJoidal . The next simple case is that of a manifold of positive or negative constant curvature, for which, that is, K r = K is constant and equal for all directions of v. It may be interest ing to note that the necessary and sufficient condition for the constancy and isotropy of curvature is *Cf. Bianchi, loc. cit., pp. 340-343. 68 Relativity and Gravitation (iX, km) =K(g tK gKp—gw gxJ, (54) for all values of the indices i, k, X, fx. These are partial differential equations of the second order for the g llc with K as a constant coefficient. They exhibit the flat manifold, K = 0, as a sub-case. Other concepts connected with the system of riemannian curvatures K v , such as the mean curvature, will be given later on in connection with gravitational problems. Here our purpose was only to show that the curvature of the four- world and, in fact, of a metrical manifold of any number of dimensions, is a concept as definite and essentially as simple as that of an ordinary surface. The only difference is that of a possible anisotropy of K v for three- and more dimensional manifolds, whereas there is, of course, no such possibility in the case of a surface. We are now ready to explain the use made by Einstein of the Riemann-Christoffel or the curvature-tensor in constructing his gravitational field equations. These will occupy our attention in the next chapter. CHAPTER IV. The Gravitational Field-equations, and the Tensor of Matter. 25. As was pointed out on several occasions, the funda- mental, metrical tensor g lK of the world determines, through the line-element ds 2 = g lK dx,dx K , its null-lines and its geodesies, and these, in virtue of the explained concrete representation, rule the propagation of light and the motion of a free particle, respectively. It remains to build up generally covariant laws or equations which would enable us to determine the metrical tensor g lK itself. Needless to say that in looking after such equations Einstein had in view, from the outset, the gravita- tional field. Of this he knew that to a certain degree of approximation it was represented by the (non-covariant) differential equation of Laplace-Poisson for the ordinary- potential, V 2 ft=-47rp, the gradient of the potential 12 giving the right hand member of Newton's (approximately valid) equations of motion. The equation of Laplace-Poisson being of the second order it was natural to look for a tensor containing the second deriva- tives of the metrical tensor components together with the g lK themselves and their first derivatives. Such a tensor lay ready in the treasury of the geometry of w-dimensional spaces since the time of Riemann, and it represented, moreover, certain intrinsic properties of any such metrical manifold, its curvature properties. This was the covariant tensor of the four-index symbols (tju. Xk) or the associated mixed Riemann-Christoffel tensor B^ x . It was natural, therefore, and Einstein himself relates to us that such was his first thought, to utilize for the purpose in hand this very tensor. 69 70 Relativity and Gravitation As we saw in the last chapter, the vanishing of this tensor expressed a simple and at the same time a profound feature of a metrical manifold, to wit, the ~ 2 n z (n 2 — 1) independent equations formed the sufficient and necessary condition for the flatness of the manifold. In our case twenty such equations form the necessary and sufficient condition for a world-domain to be essentially galilean, i.e., for its line-element to be holo- nomously transformable into c 2 dt 2 —dx 2 —dy 2 —dz 2 . A domain, therefore, in which there is no gravitational field, i.e., no premanent field of acceleration, will certainly be characterized by the generally covariant equations -B t " x = 0. The same equations might at first suggest themselves for the description of a gravitational field outside of matter. It will be seen, however, after a moment's reflection that they would be too stringent for such purposes. In fact, the field of acceleration surrounding the sun, say, can certainly not be transformed away holonomously. The said equations would thus be too stringent for such fields and, in fact, for any acceleration field which, in our nomenclature, is a permanent field, i.e., not to be got rid of by any holonomous transfor- mations of the coordinates. At the same time, the g LK to be determined are only ten in number, forming a symmetrical tensor of rank two, while the Riemann-Christoffel tensor is of rank four, and consists of twenty independent components. Such considerations led Einstein to require for the gravita- tional field outside of matter a set of broader equations, yet of the second order, and ten in number. For this purpose the symmetrical tensor derived from 2? t " x by contraction with respect to a, X naturally suggested itself. In fine, writing B" Ka = G LK , Einstein's field equations outside of matter are G lK = 0. (111°) That G lK , obtained by contraction from the mixed tensor of rank four, is itself a covariant tensor of rank two, we know Field Eoi \n" from the preceding chapter. Moreover, by (49), to be con- tracted wiili respecl to X, a, we have (56) or, alter some simple transformations which can here be omitted. c « - \ e M . ; _ i7. \ . / + 5 - • 5 2 logV — gr j tK - "I dlog V — g o,, — dx t dx. (55a) now, | i ; so that S lK (which is not a tensor) /"la/' is symmetrical, and such being also the first two terms in (55a), G lK is seen to be a symmetrical covariant tensor, of rank two; G lK = G KL . Thus Einstein's field equations (III ), valid outside of matter, are ten in number, and such is exactly the number of the metrical tensor components g u . The field equations would then give us a system of ten differential equations of the second order for ten unknown functions g„ of the co- ordinates. As a matter of fact, however, there exist between the covariant derivatives G^ of the G lK and the derivatives dG/dx t of the invariant G = g K C7,,. four identical relations (based upon certain identical differential relations discovered by Rianchi), to wit G=^G^ = ^^-,l=L 2, 3, 4. (56) d.Y, Owing to these four identities, to which we shall have to return later on, only six of the field equations are mutually independent, leaving therefore four of the g^ or any four functions of the g w free or undetermined. Such, however, should from the general relativist ic standpoint be the ease. r - d OX, 72 Relativity and Gravitation In fact, from this point of view one would expect beforehand the field equations or any differential laws to be such as to leave us a perfectly free choice of the system of coordiantes. Einstein himself, for instance, makes use of this freedom by putting in most of his formulae V — g=l, which, by (55a), reduces his field equations to -{:}+{';}{rH <*> and leaves him still a threefold freedom of choice. The latter can often be used with advantage by making gu = g2i = gzi = 0. It will be kept in mind, however, that the equations, such as (57), thus simplified do not retain their form under general transformations. They are only useful as tephnical devices offering some advanatges in the treatment of special problems. The generally covariant form of the field equations is only that obtained by equating to zero the complete or general value of G lK , such as (55) or (55a). 26. In order to see the relation of Einstein's field equa- tions to the more familiar Laplace equation, let us evaluate the curvature tensor G llc for the case of a 'weak' field, i.e. differing but litle from a galilean domain.* Thus, using a quasi-cartesian system of coordinates, let the fundamental tensor differ but little from the galilean tensor g lK , i.e., as in (21), let Sue giK ~i I uc i where all the y tK are small fractions. Then the products of the Christoffel symbols in (55) will be small of the second order, and the tensor in question will be reduced to G LK =±( La \- ±\ LK ). dx K I a ) dx a I a ) )nd order terms, {"}-K , ;]-"2"['.'] Here, up to second order terms, *Notice in passing that all gravitational fields known from experience are 'weak' in this sense of the word. Field Equations 73 and since g u = g n = g 3 x = — 1, g44=l, while all other "/ vanish, {7} ~ ["]•"■»■ {' 4 "}=ra- Thus, using the index i for 1, 2, 3 and summing every term in which i occurs twice over 1, 2, 3, we have the approxi- mate curvature tensor a dx ;[7]-i["]+^<K]-K]>« In the present connection the only interesting component is that corresponding to i = k = 4. This is, by (58), and on sub- stituting the values (13) for the Christoffel symbols, where V 2 = is the well-known Laplacian + — -+- — . dXi 2 dxi 2 dx-r dx/ If the field is stationary, the second and the third terms vanish and Einstein's last field equation, (744 = 0, reduces to the familiar equation of Laplace V ? g 4 4 = 0. (59) At the same time, as we saw before (p. 36), the equations of motion assume, in absence of gu, the form of Newton's equations £*-.*!, (236) dr- dxi where the potential = - — go* differing only by a constant factor from gu, again satisfies Laplace's equation. The complete contents of Newton's law of gravitation, thus far outside of matter, appear as a first approximation to Einstein's field equations and his equations of motion of a free particle. 74 Relativity and Gravitation 27. The ten field equations G lK = are valid outside of 'matter', i.e., as is expressly stated by Einstein, in such domains of space-time in which there is not only no matter in the ordinary sense of the word but also no electromagnetic field, or, in fact, no distribution of energy of any origin other than gravitational. Following Einstein's example the word 'matter' will be used to cover all such cases. This will har- monise with the property of energy already familiar to us from special relativity,* namely of possessing inertia, an amount of energy U being equivalent to an inert mass U/c 2 , which, by the law of proportionality, is also its heavy or gravitational mass. As we saw before, the role of the newtonian gravitation potential 12 is, in a first approximation, taken over by the tensor component gu multiplied by — c 2 / 2 . The vanishing of Gu was approximately equivalent to Laplace's equation V 2 J2 = which holds outside of matter. Within matter Laplace's equation is replaced in the classical theory of gravitation by the more general equation of Laplace-Poisson, V 2 r>= -4ttp, where p is the density of mass in astronomical units. f Now, since Gu reduces approximately, in a stationary field, to — ^ 2 g iA =p — V 2 fi, the idea easily suggests itself to make c- G 44 == - -i^- , (60) c ? and to consider this as the equation or at least as one of the field equations within matter. But, needless to say, such a single equation would riot by itself serve any relativistic purpose. What is required is a system of ten equations, of *And partly even from pre-relativistic considerations, such as in Mosengeil's investigations on an enclosure filled with radiation or those made in connection with Poynting's light-pressure experiments. fit will be kept in mind that a mass in in astronomical units is denned m 2 . by = force, so that its dimensions are \m] = [length X (velocity) 2 ]. Tensor or Mattkk which this should be one. In oilier words, fix- tensor '/,. In- to be made equal or proportional to ;i symmetrical co- variant tensor of rank two somehow associated with 'matter' and having for its 44-component the density p or what ap- proximately reduces to the usual mass density and therefore, apart from a constant factor, to energy density. Now, such a tensor was familiar from the special relativity theory under the name of stress-energy tensor often abbreviated to energy tensor. The merit of having introduced this concept into modern physics is chiefly due to Minkowski and Laue, pre- ceded in non-ielativistic physics by Max Abraham. The energy tensor made its first appearance in electromagnetism, in connection with the ponderomotive properties of an electro- magnetic field,* as the symmetrical array or matrix 5 /ll /l2 /lS Pi fn'fnfu P2 JSl fai fi3 Pi p\ pi p-i-u = / p p — u consisting of the six components /,& =fki of the maxwellian electromagnetic stress, of twice the three components pi of electromagnetic momentum (or Poynting's energy flux) and of the density u of electromagnetic energy. The physical significance of this tensor or matrix was that its product into the operational matrix d d jL a dx dy dz dt P per unit volume and its lor gave the ponderomotive force activity Pv, -lor£= \P U P 2 ,P Z , Pv|. Later on its role was generalized for a stress, momentum and energy density of any origin, not necessarily electromagnetic, *Cf. Theory of Relativity, Macmillan, 1914, Chap. IX, especially p. 23S. In reproducing it here, with pi written for gj , I drop the imaginary unit and put c = 1. 76 Relativity and Gravitation provided only that the force and its activity could be repre- sented in the form P= — V/— — , Pv = ■ — divp. dt dt From the special relativistic standpoint this array of ap- parently heterogeneous physical magnitudes was important as it transformed from one inertial system 5 to another S' as a whole, to wit by the operator A()A, where A is the funda- mental Lorentz transformation matrix of 4X4 elements and A the transposed of A. The developed form of the trans- formation equations of stress-momentum-energy need not detain us here.* The important thing in our present connection is that the said stress-momentum-energy array is a symmetrical tensor of rank two. And since such also is the contracted curvature tensor G^ , the idea naturally suggests itself to make G LK pro- portional to a symmetrical covariant tensor T lK , of which the first nine components 7n, T&, . . . T ss are of the nature of stress or equivalent to it, the components T i{ (*=i, 2, 3) replace the momentum, and the last component Ta is, or approxi- mately reduces to, an energy- or mass-density. But then it is by no means necessary (nor is it possible) to fix beforehand the exact physical meaning of the several components of such an energy tensor or tensor of matter (as it is often called by Einstein). Their significance has to be fixed a posteriori, through physical applications of the field equations aimed at. If T tK is a covariant tensor of rank two, then, as we already know, T=g LK r„ (6i) is a scalar, the invariant of T LK .f Such being the case, g lK T is again a symmetrical covariant tensor. Now, guided partly by guesses (originally at least) and partly by considerations *It will be found on p. 236 of my book quoted above. tSuch also was Laue's 'scalar' in relation to his Welttensor', i.e. the matrix .S. Field Eqi tnoNS 77 of conservation of energy and oi momentum, Einstein wrote down as his general field-equations <L = - 5e. a\ K -\ gui d, in c- the factor — S71-, c 2 being so chosen as to give, in a firsl approxi- mation, the equation of Laplace- Poisson. In fact, as we shall Bee from the more definite form to be given pre- sently, in a first approximation, r.,., = 7'=p, so that (he last of (111) gives Cm p, as in (60). Einstein's own c 2 coefficient differs from ours by the gravitation constant which i- here in- corporated into p, the density in astronomical units. The previous equations (111°), holding- outside of matter, are a special case of these general equations, for T lK = (), when also r=o. To be exact, Einstein speaks first of ' matter' as 'everything except the gravitation held' (he. cit., p. 802) and writes G lK =0 outside of matter in this sense of the word. But later on (p. 808), trying to justify the exact form (III) of his general equations, he states expressly that 'the energy of the gravita- tion field' (if there is such a thing) has also to 'act gravita- lionally as every energy of any other kind', in short that gravitation energy too has mass and weight. Thus, rigorously speaking, there is 'matter' everywhere, and the equations (111°) are valid nowhere, unless there is no gravitation field, when they are superfluous. In other words, gravitation itself contributes also to the tensor 1\ K . Its contribution, however, is practically evanescent, and this circumstance makes the equations (111°) physically applicable. But even the contribution to 7^ (», fc=i, 2, 3) of stresses within matter in the ordinary sense of the word (tensions or pressures) is practically negligible, and so is the contribution to 7^4 of the energy proper outside of molecules, atoms or electrons, and we may as well omit it in T i{ , and take, for a first approximation at least, 7'.i 4 = p, where p is the density *Principles to which we may return later on. 78 Relativity and Gravitation of ordinary matter or approximately so, always in astro- nomical units. Thus the idea easily suggests itself to build up the tensor T lK for a theoretically continuous body (a fluid, liquid or solid) out of its local density and the velocity com- ponents of its motion. For although the gravitational effects of the motion of matter are exceedingly small, yet the mere desire of writing generally covariant equations, say, of hydro- dynamics,* prevents us from discarding velocities in this connection. Thus, neglecting stresses, etc., let us introduce, after Einstein, the scalar or invariant p as 'the density' of doc doc doco matter, and the four-vector of velocity — - . Then p — * — - ds ds ds will be a tensor of rank two, a contra variant one, however. Construct therefore, by the principles explained in Chapter III, the associated tensor ?« = P &a &* — j 2 - , (62) ds ds which will be covariant and, manifestly, a symmetrical tensor. This is Einstein's energy-tensor or tensor of matter to be used in (III) whenever tensions, pressures, etc., are negligible. As a matter of fact, in view of the limitations of even the most accurate methods of observations now available, this particular tensor will cover, presumably for many years to come, all needs of the physicist and the astronomer with regard to gravitation. f The value of the invariant T or the scalar belonging to this tensor, which is defined by (61), follows at once. Since g lK g ia g K p =&g*ti =8 K ag K p=gaff , and g aP dx a dx p = ds 2 , we have, by (62), T = p, (62a) that is to say, the scalar of the tensor in question is the density of matter. *And this was undoubtedly one of the reasons by which Einstein was influenced. fThe case of hydrodynamics will be covered by subtracting from (62) the tensor pg iK , where p is the (invariant) hydrostatic pressure. 1'iKi.i) Equations 79 On the other hand, in a local rest-system, in which only (dxn/ds)' 2 survives and is equal to l/gu, we have *« - P » *'.<?., for instance, r 4 4 = p£44, and therefore, by (III), rigorously, G44 — T- gi\ P- In a first approximation (gif=rl) this gives (60) or the Laplace- Poisson equation, as announced before. The general field equations (III) may, independently of (02), be given a slightly different form. Multiply both sides, innerly, by g lK , and write G=fG„. (63) Then, since g lK g lK = 4, G = —T. (64) c- Substitute this value of T into (III). Then G lK -\ g%K G=- ifr., (Ilia) c- the required form of the field equations. Notice that C7, as defined by (63), is the invariant of the curvature tensor G iK . This invariant or rather one-sixth of it is called the mean curvature of the world, at the world-point in question. In fact, in the case of a three-space G would, apart from a mere numerical factor, be the arithmetical mean of the three principal riemannian curvatures, and this would still be the case for a manifold of any number of dimensions, at least if ds" be a definite, positive quadratic form. This justifies the name given to G above,* and equation (64), independent of the particular form (62) of T lK , teaches us that *For a certain special world to be treated later on G will be proved explicitly to be six times the smallest value of the (constant and isotropic"! curvature of three-space which it is possible to choose as a section of that four-world. (Cf. Appendix, A.) 80 Relativity and Gravitation this mean curvature is proportional to the scalar of the tensor of matter and vanishes, therefore, outside of matter. More especially, if stresses, etc., be negligible, the tensor (62) comes to its right and we have, by (62a) and (64), G = — p. (62&) c- The mean curvature of the world is thus proportional to the density of matter. Notice that p/e 2 has the dimensions of a reciprocal area, and such also are the dimensions of G, and of all G LK , since the g lK are dimensionless, and the G LK are linear in the second derivatives of the g lK with respect to the coordinates, each of which is a length. The same remarks hold good with respect to the field equations (III). It may be interesting to notice, even at this stage, that the mean curvature in familiar matter, say, in water under normal conditions, is, comparatively speaking, not insignifi- cant. In fact, remembering that the gravitation constant is 6*658. 10~ 8 , in c.g.s. units, we have for water at normal density G= 8t 6-658.1(r 8 =r86.1(r 27 cm.- 2 9.10 20 What is technically called the world-curvature is one-sixth of this. Thus the radius of mean curvature defined by R= v 6/trwill be, for water, i? = 5-688.10 13 cm., i.e., about 570 million kilometers or only 3'8 astronomical length units. But it would be rash to conclude with Eddington* that a globe of water of about this radius 'and no larger, could exist'. In fact, what is known from geometry is only that the total length of every straight (closed) line in a three-space of constant and isotropic curvature 1/i? 2 , of the properly elliptic or polar kind, is ttR, so that the greatest distance possible in such a space is %ttR and the total volume of the space is tt-R 3 . But, unlike such *A S. Eddington, Space Time and Gravitation, Cambridge, 1920, p. 148. ( i RVAT1 RE [NVAR] \m 81 a space, the world lias a non-definite fundamental differential form, and its riemannian curvature depends upon the orientation of the geodesu surface clement. Thus a direct transfer of the properties of an ellipti- space upon the (watery) world is certainly illegitimate. Notice, mon o that G, and therefore R, is remarkable as a genuine invariant of the four- world and not of a three-space laid across it as one of an infiiiii y ol possible sections. The best plan for the present is, therefore, to see in it only such an invariant of space-time, within the world-tube of a mass of water. The few numbers were here presented only to give an idea of the ordi magnitude attainable by the curvature invariant in ordinary matter, con- sidered as continuous. To those who like to contemplate sensational result 9 the best opportunity is perhaps afforded by the atomic nuclei. According to Rutherford the radius of the nucleus of the hydrogen atom is about one- two thousandth of that of the electron, i.e, U0- 16 cm., and its mass practi- cally equal to that of the whole atom. This would give for the density of mass a value 3.10 24 times the normal water density, and therefore a curva- ture radius within the nucleus V 3 . 10 12 smaller, i.e., R=82 cm. only! The moral would then be that nuclei of about this radius and no larger could exist, with the same density. Fortunately they are believed to be much smaller. But it is time to return to Einstein's equations ol the gravitational field in order to see some of their further pro- perties. 28. Multiply the field equations (Ilia), identical with (III), by g"°. Then, denoting by G t a and T" the mixed tensors associated with G lK and T lK , i.e., writing G a = o Ka G etc and remembering that g*° £„ = g?=*8* , we shall have, with 8tt K = c 2 G?- : k8?G=-KT* If G" x be the covariant derivative of G", and similarly for the energy tensor, we have, contracting with respect to X = a, ami remembering the meaning of 5", t« _ i s « dG „ ^ dG n dx a d.\\ But, by (56), the right hand member vanishes identically. 82 Relativity and Gravitation Thus, as a consequence of the field equations we have the four equations 7^=0 (' = 1.2.3,4). ( 6 5) concerning the energy tensor or the tensor of matter. Thus also, out of the ten field equations only six are left for the determination of the potentials g lK , as announced before. The matter-equations, so to call (65), as a consequence of the field equations constitute a most remarkable result.* Notice that they are entirely independent of any special form of the energy tensor, such as (62). They are manifestly general, i.e., valid for any covariant tensor T lK , merely in virtue of putting it equal, or proportional, to the curvature tensor G lK — 2 g tK G, the left hand member of Einstein's equations. In order to see the significance of the equations (65) remember that T? a is the contracted covariant derivative of the tensor T a = o Ka T Thus, if pressures, etc., are neglected, when T lK has the value (62), we have, remembering that g Ka g ia g K p =£ g K p =g lft , rpa _ dxp dx a as as More generally, if we add to (62) a tensor p llc due to stresses and any agents other than molar motion of matter, we shall have t:=p:+ p ^P~ ^-, (66) ds as where p? = g Ka p lK is the associated supplementary tensor of matter, and *Of course, the left hand members of the field equations were chosen by Einstein so as to yield these four equations of matter, as can be seen by comparing his earlier paper, Berlin Academy, 1915, pp. 778, 799, with a later, improved paper, ibidem, p. 844. Matter- Equations the covariant conjugate of the contravariant vector dx a . Of the mixed tensor (66) we have to take the covariant derivative, as defined in (42a), and to contract it with resjx < I to o and the new index. Thus the equations (65) will be ■L ta dx a But, as can easily be proved, W^+t a J 7 '" - {?}- (67) where g is the determinant of the metrical tensor. Thus, the four equations (65) become, in any system of coordinates, a Vp dx Wg Tt) - J l *\ T$ = 0, fr=i, 2, 3, 4) (65a) where the mixed tensor of matter T* is as in (66). The left hand member of (65a) is a four- vector ■*■ t to which is called the divergence of the mixed tensor T?. The equations (65a) can thus be read technically: The divergence of the mixed tensor of matter vanishes. This, of course, does not enlighten us as to their signifi - cance. To see their physical meaning take any coordinate system for which g— — 1, so that 1Z " \< (656) and consider the case of a weak gravitational field, for which the g lt differ but little from the galilean values, i.e., in quasi- cartesian coordinates, g n = — l+7n, etc., as in (21). In the expressions (66) for the energy tensor itself the y lK can be neglected altogether, so that 84 Relativity axd Gravitation d% t = — dx L (»=i, 2, 3), d£ A = dxi , and Tl=p1-p-^ — a , (f=l,2,3) 7- o . a i dx 4 dx a li =pl +p — — . as ds In the right hand member of (656) the 7 te cannot be disre- garded without anihilating that member altogether. For the Christoffel symbols vanish for constant g lK . But since their values are taken to be small of the first order, it is enough to retain in the right hand member only T i * = pf+ P (^y=Pf+ P , and since { l | ) == [ l | ] = f-^- , the four equations become, dx t if pi be negligible in presence of p, dx a ' dx, For small velocities, ds = dxi = cdt, and if ^-be the cartesian velocity components dxjdt, TJ=p^- - pVl \ Tf = p{--±p Vl Vo, etc.; Tf^pf- 9 —, etc., C 2 ' c 1 c and 7V=^+^, Tt=tf+ PV °',etc; T£=p£+p. c c Thus, neglecting cpf and cpS in presence of the momentum (per unit volume) pV of molar motion of matter, the first three equations will be (pvr — c' 1 pi 1 )+ — (pviv 2 -c 2 pi 2 ) +... + — (pvi)=p — ,etc, dxi dx 2 dt dx where 12= — \c-g^ plays again the part of the newtonian Mai rER-EQC \ i [< jxitrnti.il. tml the fourth equation ? (p*) + • »+ d( L=\ P d < /i.v, a.v 2 dx 3 dt dt or, in obvious three-dimensional vector notation. dp P d/ t- dt ami. with fi written for the three- vector c*^ 1 , pf. d i \ . d d , \ i i- * - (pwi)-f- (pfr)-f- ' (pfir.^ =(hvf ; — p — ,etc. c</ dxi dx-2 The left hand member of the last written equation is equal to — (pr^ -: ■- {pz'i) +...+: -.-: .':': V dt dx x d v_i j- = {pVi)-rpv< . <:.: V dt or. it" or be the volume and 5w = p<5r the mass ot an individual , . . dfoiSm) „. .. , . element ot matter, equal to . Similarlv we nave dt . St dp . ,. , , dp , ,. d(8m) 6/ dt dt . 8~ Ultimately, therefore, the approximate equations of matter are. with i. j, k as unit vectors along the coordinate a\<. - (y8tfi)-8r\idivf^} dwt t +kdivU]=im . Yi: A dt and d . N 99 8ni .... dt dt The first three equations embodied in the vector formula A are the equations of motion of a continuous medium* under internal stresses (tensions) fa =<Ppi, and under the action of *A deformable solid, liquid or Quid. 86 Relativity and Gravitation the gravitational field of which the newtonian potential is again, as in the case of the approximate equations of motion of a particle, ft = — \c l g^. The fourth equation, (B), is, apart from the new term on the right hand, the familiar equation of continuity. In other words, the first three equations express the principle of momentum, — the amount of momentum acquired by matter from the field per unit time being given by 8m . grad ft which is the newtonian force on the mass element 8m. And the fourth equation expresses the principle of energy or, equivalently, of matter, — the amount of energy (c 2 8m) acquired by a material element per unit time being equal to the mass 8m of that element multiplied by the local time- variation of the potential, i.e., approximately equal to the decrease of the potential energy of that element. It is scarcely necessary to say that this gain or loss in energy or in mass of ' matter ' placed in a gravitational field, according to the sign of dft/oV, is immeasurably small. Its discussion in this place may have only a mere academic interest. If it be neglected, (B) gives at once the usual equation of continuity, and (A) assumes the perfectly familiar form pV— i divfi— . . . =p V ft = gravitational force per unit volume. On the other hand it is interesting to notice that the equation (B) becomes, for V = 0, at once integrable and gives 8m=8m e~~ft/ c2 > where 8mo is 8m for ft = 0. A similar result followed from a gravitation theory proposed some time ago by Nordstrom (Phys. Zeits., 1912, p. 1126). Its interpretation may be left to the care of the reader. Returning once more to the rigorous equations (65a) we now see that the terms l a ff .Ti represent in general the momentum and energy (or mass) acquired by 'matter' in a gravitational field. The four equations themselves express the principles of momentum and of energy, as was made plain above on their appropriate form. I avoid purposely to call them principles of 'conservation' of momentum and of energy. For although Einstein succeeded in giving them the form* *Sitzungsberichte of Berlin Academy, vol. 42, 1916, p. 1115, where the German T and / are the above V — g T,"V —g t. RGY PWXCII d dx- [vr^(77+c)]-« in which they would deserve the name of conservation, yet the f* built up of the g* 1 ' and their first derivatives has not the character of a general tensor, but behaves so only with respect to a certain class of coordinate ins (for which g = — 1). In view of this it has not seemed necessary to quote here the values of the r* . Suffice it to say that since, unlike T* they contain only the gravitational potentials 'and their first derivatives), Einstein calls V — g t a ' the components of energy of the gravitational field ', and V —g T a those of matter, and reads the last set of equations: the total momentum and the total energy- of matter and of the field are conserved. The point under consideration is after all but a formal one, and we prefer therefore to content ourselves with the original equations (65ft), interpreting their second terms as momentum and energy gained (or lost) without attempting o locate them as such in the gravitational field before their passage to, or rather appearance in, matter. Historically, the position is this. In the special or restricted theory of relativity the principles of conservation of momentum and of energy- were expressed by the vanishing of the 'lor' or, in Laue's nomenclature, of the Divergence of a world tensor, this 'Divergence' being a four-vector whose components were transformed by the Lorentz transformation, the four equations themselves being thus invariant with respect to this kind of transformation. The tendency to imitate these principles of conservation in the generalized theory was but a most natural one. But the proper generalisation of that special Divergence in a theory- admitting any trans- formations of the coordinates is the Divergence denned in the general tensor calculus, i.e., the contracted covariant derivative of the mixed tensor of matter T*. This is a genuine four-vector, a covariant tensor of rank one, and the original generally covariant equations T a = 0, ' ■a are the only appropriate expression of the principles of momentum and energy. Their expanded form is (65a] and this cannot in general be given the form of 'conservation laws'. Only for constant g^ , that is in a ga'.. domain, does it reduce to d r— T« = dx a • which is identical with the vanishing of what was called the Divergence of T* in the restricted relativity theory. All attempts to squeeze the brc Divergence T a into the narrower one seem artificial and useless. For conservation as an integral law, cf. Einstein, Berlin Sitsungsber., 191$. p. 44$. 88 Relativity and Gravitation 29. That the gravitational field equations (together with the equations of motion and those of the electromagnetic field) can be deduced from a single variational principle or, as it is called, a Hamiltonian Principle, was first shown by H'. A. Lorentz (Amsterdam Academy publication for 1915-16) and by D. Hilbert (Gottinger Nachrichten, 1915, No. 3), and later on by Einstein himself.* More recently Hilbert, Weyl and others have returned to this subject in a large number of publications, in some of which the importance of the Hamil- tonian principle seems to be unduly overestimated. Since this matter is, after all, of a purely formal nature, it will be enough to give here but a very brief account of Einstein's own treatment as developed in the paper just quoted. With dx as a short symbol for dx\ dx 2 dx s dx± Einstein writes the Hamiltonian principle V^g(G+M)dx = 0, (68) where G, M are invariants. Since V — gdx, the volume of a world-element, is invariant, so also is the whole integrand. Let M be a function of the g^ and of q ( and dqjdx a , where q t are some space-time functions describing 'matter', while G is assumed to be linear in d 2 g M "/dx a dx$ with coefficients de- pending only upon the g M „. Then, by partial integration. V g Gdx = Fda, Vg G*dx + where da is an element of the boundary of the world-domain dx (the particular value of the integrand F being irrelevant), and G* is a function of the g M " and their first derivatives only. Let it be required that the values of g"" and of their first derivatives should be fixed at the boundary. Then 5 Fd<r= 0, and we can write, instead of (68), *A Einstein, Hamiltonsches Prinzip und allgemeine Relativitatstheorie, Sitzungsberichte der Akad. der Weiss., Berlin, vol. XLII, 1916, p. 1111- 1116. I Iamii.iom W PRIM III I. a \Vg (G*+M)dx~0, where the whole integrand depends upon the p/^and j, and i heir first derivatives only. Tims the variation ol the ■■" gives at once i he ten equal ions where and d dH* dH* dN_ dx a dp( a ) dp dp H* = V~g c*, X = V - 1 M p-r* Pw= dx a Now, let G \)v the curvature invariant, i.e., in our previous notation, Then, on performing the said partial integration, it will be found that With this value of H* the (•([nations (69) become identical with Einstein's field equations as given above, if we put f = —V—p T i.e., „„ dN d;< = — V — P " a 1 .-. .Sep - 1 < or, in terms of the covariant tensor oi matter, \ dM _ dfff' -Tat- 71) The verification of this statement may be left to the care of the reader who may confine himself to systems for which 90 Relativity and Gravitation 30. Gravitational waves. Let us close this chapter by briefly mentioning a method of approximate integration of the field-equations given by Einstein (Berlin Academy pro- ceedings for 1916, p. 688) which exhibits the propagation of gravitational disturbances. Let the g lK differ but little from the galilean values, in a cartesian system, say, or — in our previous notation— let where the y tK are small. Then Einstein's approximate solution of his^field-equations is 7„=T , «-|5:7 / xx, (72a) where y\ K is the retarded potential of — 2k T lK , that is to say, the familiar particular solution r ^K I IK ~. 4-7T J — T lK (x , y , z, ct — r) dx dy dz (72b) r of ' the wave-equation ' G-£-'*) y -— * r - (72c) In (72b) r is the three-dimensional distance of the point for which y\ K is required (for the instant /) from the integration element dxdydz at which the value of T lK prevailing at the r . instant t — — is to be taken. c This solution represents gravitation as being propagated with the normal light velocity c, the slight changes of the latter due to the gravitational field itself being manifestly neglected. In this approximation the rigorously non-linear field equations are replaced by linear differential equations of the form (72c), the usual wave-equations. In the sub-case of a stationary gravitational field, when the whole tensor of matter is reduced to T Ai = p, we have by (726), as the only surviving y' lK , , k pdxdydz _ kS2 744 ~"2x J J J "~r " 27' Approximate Integration (, i where Q is the ordinary newtonian potential of the gravitating masses, and, by (72a), the only surviving y lK , Kit 212 711 = 722 = 733= —744= — - — = — —i 47T C 1 so that, as before, the role of the potential ft is taken over in part by — %c 7 y u - —7 CHAPTER V. Radially Symmetric Field. Perihelion Motion, Bending of Rays, and Spectrum Shift. 31. In order to represent the motion around the sun of a planet as a 'free particle', of mass negligible compared to that of the central body, it is enough to find a radially sym- metrical solution of Einstein's field equations outside the sun, G lK = 0, (111°) considering the origin r = of polar coordinates r,<f>, 6 as a singular point. As a form of the line-element, sufficiently general for this purpose, let us assume ds%)= gi dr 2 ^ r-dtf - r 2 sin 2 </> dd 2 + g± cWt\ (73) where g\, gi, written instead of gu, gu, are functions of r alone, of which we shall thus far assume only that ki(») = -li «*(») = !, (74) i.e., that at distances r large compared with a certain length belonging to the sun (which will appear in the sequel) the line-element tends to its galilean form ds 2 = —dr 2 — r 2 (dcjr + sinW0 2 )+cW. Let us correlate the indices of the coordinates by putting x h x 2 , x z ,Xi = r, 4>, d, ct respectively. Then the metrical tensor in question will con- sist of the components gi = giO) , g 2 = - r 2 , g 3 =- r 2 sin 2 <£, g 4 = g*(r) , (73a) where g K has been written for g KK . In the more general case ds 2 = g K dx 2 , in which the g K = g KK are any functions of all the variables, we have, for the only surviving associated tensor components, 92 Radially Stmmei ri< Field „ kk = and i herefore, recalling the definition of the Christoffel symbols, k ) 2g K Bx t 2& dx. for all t, >. , for i=fc«c, while all other Christoffel symbols vanish. Applying these formulae to the more special tensor (73a), writing h=\og gi, lu = log gt and using dashes for derivatives with respect to r=Xi, we have the rigorous values of the only surviving Christoffel symbols, altogether nine in number, (11) J12\ = (U\ _ 1 (U\_ \ll-**'\2j \3/- r > 14 (- = ** |22\ r [23\ )33\ r . 2 , /33 V ~\ -, ( — • - Slll"<P,-\ rt /■— oMIliip, -I = -§sin2«£; . J44\- N-i These values substituted into the general expressions (55) for the components G lK of the curvature tensor give zeros for all those having lt^k, while the remaining four diagonal com- ponents are, rigorously, r 6^33 = sin 2 </> . GU G 44 = H feu + T (// 1 '+// 4 ')~1 . gt 1_ r J (77) 94 Relativity and Gravitation Thus we have, according to the field equations for r>0, that is, outside of matter, for the two unknown functions g\, gi the three differential equations OZ, ' (a) A/'+iVfo'-Ai')- — =0 r (b) £(J£'-Ai')+fi+l = (c) hi'+h4' = 0. The last of these equations gives /?i+/f 4 = log(gi g 4 ) =const., that is to say, by (74) , gigi = const. = — 1. Equation (b) now becomes gi-{-rgA=l or -f log[r(g 4 -l)] = 0, dr so that r(gi— 1) = const. = — 2L, say. Thus the rigorous, and the most general, solution of the field equations (b), (c) is ultimately, 2£ /\ 2LN- 1 , gi= -Ql- — y , (78) where L is any constant, some length, characterising the sun, i.e., here the singular point or centre of the gravitation field. As to the first field equation (a) it is satisfied identically by these values of gi, g 4 -* To express the constant L in terms of M, the sun's mass in astronomical units, we may apply the following reasoning: As we already know, in the approximate equations the c 2 newtonian potential Sl = M/r is replaced by 744. Now, *In fact, since gigi= — 1, the left hand member of equation (a) becomes hi"+hi' 2 +2hi'/r, and this is, by (78), -^frTEF [u-*w-««l which vanishes identically for all r=t=2Z,. 1=1' ~ r Radially Syhmetrn Field in the present case, 744 = £'i — 1 = —2L/r. Thus .\f=< I.. whence l= M . n c* Ultimately therefore, the line-element (73) corresponding to a radially symmetrical field becomes, rigorously, ds i = (l- — \ftft"-M -—Xlr-r^dtf+sm^dd 2 ), (80) a form of the solution of the field equations first given by Schwarzschild (Berlin Academy proceedings, 1916, p. 189). As was already mentioned in Chapter IV, the dimensions of L as defined by (79) are those of a length. This length, which is sometimes called the gravitation radius of the given body, amounts for our sun to about 1*47 kilometers. Thus, for all applications of any actual interest, L/r is a small fraction and the coefficient of dr 2 can be replaced by — (1 + 2L/7). 32. Perihelion motion. Let us now consider the motion of a free particle (planet) in the field determined by the line- element (80), that is to say, by the metrical tensor ft = - ( 1- —I, & = -r 2 , &= -r-sin-</>, g 4 = 1 j The general equations of motion (15) with the values (76) of the Christoffel symbols become, for t = 2, 3, 4,* d°-4> _ ds* ~ 2 r dr ds d<t> ds d-d _ d?~ -K dr ds +c d 2 Xi ~d? --W dr ds + isin2*g) ; (fa J </s *Instead of the first equation of motion (1 = 1) it will be more convenient to take the identical equation g lK &, X K = 1. 96 Relativity and Gravitation Lay the plane $ = 71-/2, the equatorial plane of the coordi- nate system, through the direction of motion of the planet at some instant / - Then, at that instant, d4>/ds = § and sin 2<f> = 0, and therefore, by the first equation, permanently = 71-/2, that is to say, the planet will describe a plane orbit, and the remaining two equations, together with the identical equation giK^iX K = l, will become 6 + — 0=0, r x 4+/Z4' rxi = 0, g iXi *- -^-1*^=1, (80a) Si where /* 4 = log g 4 and g 4 =l — 2L/r. The first two of these equations can be written — log (r 2 e) = 0, — log (g 4 *<) = 0, as as , and give r 2 'd = p, giX4 = k, (81) where p and k are arbitrary constants.* With the values of ±i and 6 derived from (81) equation (80a) becomes ' r 4 or, putting p :© ! +(^)0-f) (±y. fe2_i or + — p-p 2 +2L P 3 . (82) £2 £2 The determination of the orbit is thus reduced to a quad- rature. As an alternative we may write down the differential equation of the orbit, by differentiating the last equation with respect to 0, *The first of (81) represents the slightly modified law of Kepler: areas swept out by the radius vector in equal proper times of the particle (s/c instead of t) are equal. Perihelion Moth P-+p=- L - +ZL,r. (82a, der- p- Hither equation differs from the familiar equations of celestial mechanics, based on Newton's principles, only by the under- lined last term of the right-hand member. It is well known that in the absence of this supplemental^- term the orbit is a conic (an ellipse, a parabola or a hypert p = — [l + ecos(0-«;] (83; P 1 with fixed perihelion. w = const. In fact, equation (82j identically satisfied by (83) ; and so is (82) if we put ^(l-e-j=L-(l-fe 2 ), (83a) so that the orbit is an ellipse, a parabola or a hyperbola according as k- is smaller, equal to or greater than 1. In general, for orbital velocities comparable with the light velocity, equation (82) gives d as an elliptic integral of p. to which corresponds a complicated non-closed orbit. Its dis- cussion may be left to the care of the reader.* Here it will be enough to consider small velocities such as occur among the planets of the solar system. The supplementary term is then small compared with the newtonian ones, and the problem can be solved approximately by a conic (83) with slowly moving perihelion. If dp dd is the derivative of p when u is kept fixed, and if the term with (do: dd)- is neglected, we have (dp V _ /dp_ dp dwX 2 _^_ /dp V i dd ) \dd ' du dd ) ~ \dd ) dp dp do: <dd / \dd da dd / ' \dd / dd do~: dd and since (dp dd)- itself accounts for the first three terms of the right hand member of (82), the perihelion motion will be determined bvt *Cf. A. R. Forsyth, Proc. Roy. Soc, XCVI1 1920 , p. Ho. also \V. B. Morton, Phil. Mag., XLII (1921), p. 511. tThis reasoning, aiming at the seculcr motion of the perihelion, pre- supposes the knowledge of absence of a secular variation of the eccentricity €. Cf. footnote on p. 99, infra. 98 Relativity and Gravitation dp dp du _ 3 89 du> dd Here (83) can be used with sufficient accuracy for p and its two derivatives, so that dco _ L 2 l+3ecos w+3e 2 cos 2 «+e 3 cos% dd ]&? sin 2 « where u = d — w. Integrating this from to 2-k over 6 or, what for our approximation is the same thing, over u, we shall have the secular perturbation 8u>, the motion of the perihelion per period of revolution. The second and the third terms of the integrand, having in the second and the third quadrants values opposite to those in the first and fourth, contribute nothing to the secular perihelion motion, and the same is true of the first term, since this is the derivative of the periodic function — cot u. We are thus left with K~ 3L2 p* J cot 2 « du, and since — cot 2 w is the derivative of cot u-\-u, 5u=?^l. (84) P' 2 This being essentially positive, the secular motion of the perihelion is progressive, that is, in the sense of the revolution of the planet. If the orbit be an ellipse (e 2 <l) with semi-axes a, b, we have, by the original meaning of the constant p, 2 dd _^ r 2 d9 _ 2irab ds ' c dt cT where T is the period of revolution, and by (83), ~ "c 2 " ¥ ~ c 2 T 2 ' expressing Kepler's third law. Thus Perihelion Mono K 2ira~ 'lira p cTb cTVl-e* ' and (84) becomes 24ttV , c 2 r 2 (l-e-) which is Einstein's formula for the secular motion of the perihelion of a planet, undisturbed by other planets, per period of revolution.* This formula gives for Mercury, per century, 43" or 43" "1, coinciding most remarkably with the famou- excess of perihelion motion of that planet, unaccounted for by the perturbations due to the other members of the solar family of celestial bodies. Although the rival explanation based on perturbing zodiacal matter, due to Seeliger — New- comb (taken up more recently by Harold Jeffreys), cannot be considered as ultimately discarded, this is certainly a most conspicuous achievement, perhaps the greatest triumph of Einstein's theory, yielding the required excess without the aid of any new empirical constant in addition to the light velocity and the gravitation constant. As to the remaining planets, Einstein's formula gives for them secular perihelion motions too small to be either contradicted or confirmed by observation in the present state of the astronomer's know- ledge. In fact, the only other serious anomaly unaccounted for by newtonian celestial mechanics (unless Seeliger's theory is accepted) is the excessive motion of the nodes of Venus, but with this Einstein's theory is essentially powerless to deal, since it yields, for a radially symmetric centre of course, rigorously plane orbits. But even the outstanding node motion of Venus is generally felt to be much less important than Mercury's perihelion motion yielded so naturally by Einstein's theory of gravitation. *A more thorough analysis shows that this is the only secular pertur- bation, the eccentricity, the period and the remaining elements of planetary motion being unaffected by the deviation of Einstein's theory from that of classical celestial mechanics. Cf. W. de Sitter's paper in Monthly Notices of the Roy. Astr. Soc, London, 1916, pp. 699 ct scq., more especially sec- tion 17. 100 Relativity and Gravitation 33. Deflection of light rays. The propagation of light is given by the minimal lines ds = of the metrical manifold determined by the quadratic form (80). By reasons of symmetry it is again sufficient to consider the plane (j) = const. = 7r/2. Thus the light equation becomes — dr- + r W = g A c 2 df, g 4 =l-—. If v be the system-velocity or the 'coordinate velocity' of light, defined bv -(JMS)Mf)'. the preceding equation gives K(f) 2 -Kf)>-- and if t\ be the angle between the tangent to the light ray and the radius vector, so that dr / da = cos rj, rdd / da = sin -q, c- 1 [~COS 2 ?7 . „ ~| , , — = — + sin 2 rj . (85) v 2 gi L gi -1 Thus, if the ray be radial, away from or towards the origin, the light velocity is cg4, and if transversal, c 'vgi, both principal velocities being smaller than c, and both tending to c at infinity. Neglecting the square and the higher powers of L/r, which even at the surface of the sun is a very small fraction, we can write, approximately, v/c = 1 — (l+cos 2 J7)L/r. The velocity of light being determined by (85), the shape of the ray or the light path between any two points 1, 2 can be found* by means of Fermat's principle dt=8 ^ =0. In fact this principle can be proved to hold, at least for stationary gravitational fields, i.e., for g lK not containing the *In terms of r, r\, and thence by integration in terms of r, d. Deflb noN 01 i< ■ 101 time* as in the case; in hand. Those interested in such an application of Format's principle may consult de Sitter's paper quoted in the preceding section. But a much more speedy way of obtaining the light path is to consider it as the limiting case of the orbit of a fret- particle. In fact, returning to the differential equation (82a; of such an orbit, and remembering the original meaning of the integration constant p, _ 2 </0 ds we have for light, or for a 'particle' which would everywhere keep pace with it, p= co, so that the differential equation of the light path becomes ^ +p-3Lp 2 = 0. (86) dd- In the absence of the last term, which bears to p the small ratio SL/r, we should have p = po cos 6, a straight line whose shortest distance from the origin is r = 1/po, the angle being measured from the corresponding radius vector. Thus, replacing p in the last term by p cos 0, which amounts to neglecting IJp-- and higher order terms, we have for the light ray p = po [cos 0-+- Lpo(l +sin 2 0)J or — =cos0+ — (l+sin-0). r r The angle between the asymptotes (r/ro= °°) of this curve is easily found to be A = ^ = 4 ^. (87) r c«r This is then the total angle of deflection of a light ray arriving *For a simple proof see T. Levi-Civita's paper in Nnovo Cimento, vol. XVI, 1918, p. 105. Levi-Civita assumes also gn=gu = gn = 0. The latter limitation, however, does not seem to be necessary. Thus, for instance, it can be shown that Fermat's principle leads to a correct result in the case of a uniformly rotating system, i.e., obtained from a galilean system by the transformation d' = d-{-(j)t, co = const. 102 Relativity and Gravitation from a distant source (star) to the earth, if r be, approxi- mately, the shortest distance of the ray from the origin, e.g., from the sun's centre. In the latter case, if R be the sun's radius, we have 4L/i? = 5'88/6"97.10 5 radians = 1"75, so that A=l"75- . This is Einstein's famous formula for the displacement of star images seen in comparative angular proximity to the sun's disc. It can be considered as fairly well verified by the results of the Eclipse Expedition at Sobral, Brazil,* of May 29, 1919, which were ultimately estimated to give, when reduced to r = R, the value 1"'98 with a probable error of about six per cent. This is certainly more than a mere order-of-magni- tude coincidence, and speaks strongly in favour of Einstein's theory. The displacements according to Einstein's formula should, of course, be away from the sun and purely radial. The displacements measured on the Sobral plates deviated from radial directions, at least for four out of the seven stars, considerably, to wit by 35°, 16°, 8°, and 6° for the stars numbered 11, 6, 2, and 10, whose distances from the sun's centre were about 8, 4, 2, and 5R respectively. These deviations or the presence of transversal displacement components may well be due to the distortion of the coelostat-mirror by the sun's heat, as pointed out by Prof. H. N. Russell. Yet a refined investigation of this point during the next eclipse seems very desirable, and, as I understand, will be taken special care of at the Eclipse Expedition of September 20, 1922, at which it is designed to avoid the use of a mirror. The field of stars near the sun, during totality, will then be almost as favourable as in 1919-t 34. Shift of spectrum lines. Consider an atom, say of nitrogen, placed in the photosphere of the sun, at rest or practically so. • Then its line-element or the element of its 'proper time' will be, by (80), and writing for the present 5 instead of s/c, ds — / 2L\ A *The measurements of the Principe Expedition, made under un- favourable weather conditions, seem by far less reliable. fSome preliminary details will be found in Monthly Notices of the Roy. Astr. Soc. for May 1920, p. 628. Spectrum Shim 103 and any finite interval of its proper time Let another nitrogen atom he placed in one of our terrestrial laboratories, at a distance r from the sun's centre. Then its proper-time interval will be In particular, let A/i be the terrestrial, and At the solar time period of one of the natural vibrations or spectrum lines of nitrogen. Now, encouraged by the traditional belief in the somewhat vague 'sameness' of atoms of a given kind, Einstein assumes, as he did already in other circumstances in the special rela- tivity theory, that the said two atoms are 'equal' to each other in the sense of the word that the proper time?* of their vibration periods are equal to each other. Eddington in his Report (p. 56) simply says that an atom is "a natural clock which ought to give an invariant measure of an interval 8s, i.e., the interval 5s corresponding to one vibration of the atom is always the same". Weyl states the case in an apparently more profound way by saying that if the two atoms are "objectively equal to each other, the process by which they emit waves of a spectrum line, when measured by the proper time, must have in both the same frequency". In short, the founder of the theory, as well as his exponents assume, more or less implicitly, that As = Asi. If so, then the ratio of the solar to the terrestrial period of vibrations is £-0-.f )'(-!)■ or, since in our case R/r is but a small fraction, ^L =1+L =1 + 2109.10-°. (88) Ah R *It is now usual to extend this name for ds/c from special to general relativity theory. 104 Relativity and Gravitation Einstein's conclusion then is that the lines of the solar spectrum, compared with those of a terrestrial one, should be shifted towards the red, the proportionate increment of wave- length being *h = L =2-109.1(r 6 , X R or equivalent to a Doppler effect due to a (receding) source velocity of 0'633 kilometers per second. This amounts, for violet light, to about 0'008 A. Now, although with the modern means one-thousandth of an A or even less can be well detected in comparing spectra, Dr. St. John of the Mount Wilson Observatory, who observed 43 lines of nitrogen (cyanogen) at the sun's centre, and 35 at the limb, was unable to detect any trace of the predicted effect. His observations were made and discussed in 1917, and his final conclusion then was that "there is no evidence of a displacement, either at the centre or at the limb of the sun, of the order 0'008 A". Since that time, however, in view of the entanglement of the Einstein effect with shifts of a different origin, and seeing that the results of other astrophysicists were not quite so definite, Dr. St. John suspended his final judgment and is now taking up a thorough discussion of the whole material of solar spectrum shifts from E. L. Jewell's first observations, made about 1890, up to the present. The natural impression now is that it would be premature to either assert or deny the existence of the gravitational spectrum shift. Einstein himself has, on more than one occasion, expressed the very radical opinion that, should the shift be absent, the whole theory should be abandoned. Yet, in view of the hypo- thetical nature of the sameness of atoms in the explained sense of the word, such an attitude, though personally in- telligible, is by no means necessary. It is true that the in- variability of an atomic ^-period of vibration in a gravitational field can, with the aid of the equivalence hypothesis, be re- duced to its invariability while the atom is being moved about, — a property of atoms as 'natural clocks' already Na'm ral Cloi b 106 utilised in special relativity.* Yd we do not know whether the atoms actually possess even the latter property. Thus, Einstein's intransigent attitude proves only the Btrengtl his belief that the atoms are or will turn out to he such natural, ideal clocks. But, after all, this is only a gu A very reasonable one to be sure; for if not among the atom-. then there is indeed but little hope to find such clocks among other 'mechanisms', natural or artificial. At any rate, a final astrophysical verification of Einstein'- spectrum-shift formula, supported perhaps by repeated experiments on canal rays, would be an achievement of fundamental importance. Until then 'the natural clock' will remain a purely abstract concept. *It is this theoretical attribute of atoms which has led to the conclusion that moving hydrogen atoms (canal rays) will emit, in transversal direc- tions, waves (1— v 2 /c 2 )~' / * times longer than atoms at rest. But even this shift effect, though tried experimentally, does not seem to have ever been detected. CHAPTER VI Electromagnetic Equations 35. Maxwell's equations of the electromagnetic field in empty space supplemented by the convection current pV, or the fundamental equations of the electron theory are, in three-dimensional vector notation, with x i = ct, — + curlE = 0, divM = dXi r- curl M = p — , div E = p. dx± C They contain, apart from the velocity v of moving charges, but two vectors E, M which may be provisionally called the electric and the magnetic forces. As is well-known from the special relativity theory, these equations retain their form or are covariant with respect to the Lorentz transformation, i.e., in passing from one to another inertial system.* They are not, however, generally covariant, and thus not appropriate to the purposes of the general relativity theory. What is covariant with respect to any coordinate trans- formations is the somewhat broader system of equations, containing two more vectors D and B which may be called the electric and the magnetic polarizations, f aB dXi +curlE = 0, divB = 0, (A) - — +curlM = p-, divD = p. (B) dXi c *Cf. for instance my Theory of Relativity, 1914, Chap. VIII, and, for the historical aspect of the subject, Chap. III. fOr the electric displacement and the magnetic induction respectively. 106 Electromagnets Eqi atio iot In a galilean domain or an incriial system D and B reduce to E and M respectively, bul in general, in a gravitational field or a non-incrtial system, the polarization> differ from the forces, being some linear vector functions of the latter The general covariance of these two groups of ell magnetic equations was first noticed and developed by F. Kottler as early as in 1912* and shortly afterwards, with due acknowledgement, incorporated by Einstein into the physical part of his general theory of relativity. Let F lK be an antisymmetric covariant tensor of rank two or a six-vector, which will embody in itself B and E, and thus may be called the magneto-electric six-vector. Then the group (A) of equations can be replaced by the equations dx x dx t dx K which are generally covariant since their left hand members are, by (46), Chap. Ill, the components of a general tensor of rank three, the antisymmetric expansion of the six-vector F lK . To compare (Ai) with (A) and to see the simplest form of the correlation between B, E and the six components of F lK use cartesian coordinates or, in the presence of a gravita- tional field (always 'weak'), quasi-cartesian coordinates and denote by 1, 2, 3 the rectangular components of B, E along the three axes. Then the group (A) of equations will be dB, . dE, BE* n , — + — — — =0, etc. dXi d.Yo dx 3 dxi dx-: dx 3 where 'etc.' means two more equations by cyclic permuta- tion of the suffixes 1, 2, 3 only. On the other hand, writing out (Ai) and remembering that F iK = —F a . we have *Friedrich Kottler, Raumzeitlinien der Minkowski? schen Welt, Sitzungs- berichte Akad. Wien, vol. 121, section Ho, pp. 1659-1759. 108 Relativity and Gravitation dXi dx 2 dx s dF 23 dFzi dFn =Q dXi dx 2 dx 3 and these four equations become identical with those just written if we put F 23 , Fzi, F n = Bi, B 2 , B s Fu, F 2i , Fzi — Ei, E 2 , E 3 respectively, or more compactly, if i, k be reserved for 1, 2, 3 only, F ik =B; F H = E. (89a) This then is the required correlation for the case in hand. Non-cartesian coordinates will be dealt with in the sequel. Next, let F lK be the supplement of F aP defined, as in (34), by F-^C^F^. (90) Then the group (B) of the electromagnetic equations will be replaced by the four equations Vg dX K where C is a contravariant four-vector. Such also being the left hand member, the divergence of F lK , as in (47), the equa- tions (Bi) will be generally contravariant. To compare them with (B) and to find the correlation proceed as before. Thus, on the one hand, _ dDi dM 3 _ dM 2 _ Vi dx± dx 2 dx 3 c Mh &D2 dD 3 _ dXi dx 2 dx 3 and on the other hand, remembering that F KK = and F lK = -F K \ Electromagnetic Equations 109 if —g C\ etc., d . . — (V-g F 41 ) + etc. = V -g C*. OX\ The required correlation is, therefore, V^(F<>, F<\ F«)=D U D„ A vCj (/?* F 3 \ F l -) = M 1} M 2 , M 3 or, in the previous abbreviated notation, V^7f 4 * = D; V^F ik = M. (896) Since F" is thus seen to embody the electric polarization and the magnetic force, it may be distinguished from its supple- ment by the name of the electro-magnetic six-vector. At the same time we have, by comparing the right-hand members of the two forms of equations, V~ g {c\c\c>- c*)= P (jj, v f, * ; i) or, more shortly, C; C*= -4=,( V -- 1 Y (91) exhibiting C* as the electric four-current. It is interesting to note that since we can put v i /c = dx i ,'dx i and dx A dx A =\. the last correlation can also be written C*=-4= ^f. (91') V — g d.\u Since dx K is a contravariant vector as well as the four-current, the. factor of dx * will be an invariant, and since V — g dxj dx°dx s dx 4 is also an invariant, the volume of a world-element, we see that the electric charge 8e=pdxidx>dx$ is again an invariant. Then, however, not p itself but p divided by the determinant — |gi*| will be the system-density of electricity. 110 Relativity and Gravitation It may be well to illustrate the general transformation formulae of Fin r p, _dx„ dxp F by writing them out for the simplest case of two inertial systems S, S' in uniform translational motion relatively to each other. The transformation is in this case the familiar Lorentz transformation, i.e., in cartesian co- ordinates and with the X\ axis along the direction of motion, xi = y(xi'-\-(3xi), xt = Xi', xz=x% ', xt = y(Xi-\-fixi'), where fi=v/cand y = (1— j8 2 ) ~ **, if v be the velocity of S' relatively to S. First of all, since in this case the g LK have their galilean values (in both systems), we have B=M, D=E, so that there is no need to consider the supplement of F lK ; it is enough to treat F lK itself. Next, since X2, x s depend only on x 2 , x 3 \ being equal to them respectively, we have dx a dxa F' 23 = M 1 '= r-* t-2, F afi = F 3t = Mi. 0x2 0x3 ox F'. 3 i=~ F Sa =y(F 3l +l3F Si ), OXi Similarly, M 2 ' = y((3E 3 +M 2 ), and so on. Thus we get the transformation formulae Mx'^Mi, M-/=7(M 2 +/3E 3 ), M 5 ' = y{M z -^E 2 ) E^Eu E 2 '=y(E 2 -PM s ), E 3 ' = y(E 3 +^M 2 ), familiar from the special relativity theory. The corresponding transform- ation of the four-current may be left as an exercise for the reader. It will be kept in mind that the correlations of the forces, the polarizations and the current and charge density to the two conjugated six-vectors and the four-current given in (89a), (896), (91) are valid only for the particular case of a cartesian or quasi-cartesian coordinate system. With other systems, such for instance as the polar coordinates, even in a galilean domain, the correlation formulae are more compli- cated, and contain besides the determinant g the several components g lK of the metrical tensor or (in a non-galilean domain) parts of them, as will be seen later on. It is important Electromagnetn Equations ill to understand that there is nothing general about th correlations, apart from the faet that I\ K embodies somehow (he three-veetors B and E, and F u the vector- D and M, and C K the convection current and the charge density, everything being entangled with the metrical tensor and through it also with gravitation. From the standpoint of general relativity the n equations are henceforth no more the broadened maxwellian equations (A), (B) but the set of generally covariant or contravariant equations (Ai), (BO with the metrical link (90; between the two six-vectors. It will be well to gather here these somewhat scattered equations; the whole generally covariant electromagnetic set is thus dF lK dF± f dF Xt _ Q dx x dx t dx K 1 d I / x V? OX K F' K = o La o KP F 1 t> & •* a, (IV) This will read as follows: the expansion of the magneto- electric six-vector vanishes; the divergence of the electro- magnetic six-vector, the supplement of the former, is equal to the electrical four-current. 36. The four-potential. Manifestly, the first of the equa- tions (IV) will be identically satisfied if we put F lK - d -± - *+±, (92) dx K d\\ where </> t is a covariant vector. If this be substituted, the six terms destroy themselves in pairs, and the covariant nature of (j> t ensures the required tensor character of F iK , the rotation of 4>, (cf. p. 61). The latter, which is seen to embody Maxwell's vector potential and the electrostatic potential, is called the four-potential. With the correlation (89a) the six equations (92) become 112 Relativity and Gravitation b d /d d d \ -CUrl (01, 02, 3 ), E=— (01, 02, 0a) -( t— , x—, t— 104 co7 \axi 0*2 ox 3 / dA B = curlA, E=- — -V0, coV exhibiting the three-dimensional vector A = — (0i, 02, 03) as Maxwell's vector-potential and = 04 as the electrostatic potential. The first group of equations (IV) being thus satisfied by (92), the second group gives ;HK^'(S -£')> c ' (93) which, assuming g iK to be known, are four differential equa- tions of the second order for as many components of the four- current. Since the four-potential enters only through its rotation, we can without loss to generality subject its com- ponents to a kind of solenoidal condition, as follows. If 4> K = g Ka a be the associated four-potential, a contravariant vector, then its divergence defined by (48) is a general in- variant or scalar, and the condition in question can be written £- K (Vg>)=0. (94) In a galilean domain the equations (93), (94) become d 2 A „ 9A 1 d 2 — -v 2 A= — pv, — - — -V 2 = p c 2 dt 2 c c 2 dt 2 divA+-i , il = o J c dt the familiar equations of the electron theory for the vector potential A= — (0 1; 2 , 3 ) and the electrostatic potential = 04. In general, however, the equations (93) for the four- potential will contain in a complicated way the components of the metrical tensor, which again means an entanglement of the electromagnetic with the gravitational field. This mutual relation of the two fields appears directly in the third of equations (IV) giving the general connection between the magneto-electric six-vector and its supplement. The Four-Potential 113 Since the four-potential is a covariunt and dx t a COntra- variant vector, their inner product dl = 4>«dx K is an invariant. This invariant linear differential form plays the same role with respect to electromagnetism as the quad- ratic differential form d^-g ut dx t dx K with respect to gravitation. As the latter determines, inter alias, the gravitational field, so does the former determine the electromagnetic field. This is only a different way of stating that the <j> K , the coefficients of dl, determine the electromag- netic, similarly as the g lK determine the gravitational field together with the riemannian metrical properties of space- time. Recently a differential geometry somewhat broader than Riemann's was proposed by Weyl who goes deep into the matter and attributes to the linear differential form an equally fundamental metrical (gauging) function as to the quadratic differential form. But reasons of space prevent us from entering here into this subject, and the interested reader must be referred to Weyl's own book* for further information. Moreover, these new physico-geometrical speculations, although undoubtedly attractive, are still being debated between Weyl and Einstein, f and may therefore be appro- priately omitted in a book of the present type. 37. Let us once more return to the electromagnetic equa- tions (A), (B) in order to compare them with the tensor equations (IV) for the case of a non-cartesian system of space coordinates. As a good example of this kind we may take any orthogonal curvilinear coordinates Xi, .v_>, x%. It is well known that if the space line-element in these coordinates be given by «fo»-f*g! + ^L + ** =^L (96) Wi 2 Tc'o'-' 10a 8 icr *H. Weyl, Raum-Zeit-Materie, 3rd ed., Berlin 1920, §10 anil §34. fCf. Einstein's remarks to Weyl's paper, with Weyl's reply, in Berlin Sitzungsber., 1918, and Einstein's recent paper, ibidem, 1921, pp. 201-204. 114 Relativity and Gravitation and if R. : be the components of a three-vector R tangential to the ^',-lines of the network. Ld.Vi \W2W3S 8x2 \u' 3 U'i' dx 3 VaPtfOgX—l and the curvilinear components of curl R are (curl R^w.wTAf^ _ l/*Yl etc. (98) With these expressions the group (A) of equations becomes, provided of course that the w\ are independent of time. ±(*-) + ±(h\-±.(^). . exc .. dXi \u- 2 w ?J / dx s \ Wz ' dx 3 \U' 2 ' L(Jl\ + ±(*l) + A. fA) = „, dXi \Wo "d}?y bXn \IL'3 W\* dXz \u-'l U'o ' and similarly for the group (B) of electromagnetic equations. These equations are to be compared with the first and second of the tensor equations (IV). To find the required correlation in terms of the g^ notice that if the domain is assumed to be a galilean one,* we have ds 2 = g LK dx t dx K = dx± 2 — da 2 , so that 1 - git =— — , £44=1, and the remaining g lK vanish. Under these circumstances the comparison gives at once, with g t written for g ih *Otherwise, say in the presence of gravitation, not the whole of — is m to be thrown upon the coefficient of dxi in the expression for the length ddi considered from the system-point of view. THOGON'AL COOR. 11 1 1 F*= -7= -If i. etc.; f = o = (K\ \ . etc.. = p. ■:. which is the required correlation. The relations between the polarizations and the forces. determined in general by the third of equations (IV), follow ict, since in the present case g" = =1. and the remaining g^ vanish, we fa F* : that is to s j 1 = - F m and therefore, by = B=M. D = E. the polarizations are identical with the : :ua- tior.- A B reduce the usual electromagne: :ons for the vacuum, giving - j> it veloci: - result might have been expected from that correspor. ling - - by the use of curvilinear ins: ; - - 38. Let us now consider the relation be B. D and M. E in another example which, bes es g " in a general way. will show bow mag: - as is If a system be.- .=..= . dimensional line-element can be is 8 = ■ - In a weak g . . - but litt!. 2 ? " . - . 116 Relativity and Gravitation cartesian or quasi-cartesian coordinates, gu and the ga will differ but little from + 1 and — 1 respectively, and the remain- ing g ik will be small fractions. Thus, from the system-point of view,* the electromagnetic equations (A), (B) will be , «2i + ™1 _ ^=0,etc, d%i dx 2 dX 3 so that a comparison with the tensor equations will give again F ik =B ) F ii = E;F ik =- 7 =m, /*= "7=D , (89) v-g V^-g as in (89a), (896). Since g 4 ; = 0, the general relation F lK = g ia g K p F afi between the two six-vectors will now give F 23 = F 23 (g 2 2 g33 ~ gj + ^ 31 (g 2 3 go. - &1 g3 3 ) + ^ (g21 gS2 ~ gll g3l) and two similar equations for F 3h F i2 . But these are the solu- tions of the three equations F 2S = — (g n F 2S -\-gi 2 F 31 -\-gi 3 F 12 ), etc., h where h is the determinant | g ik | . Now, h = g/gn, and therefore F 2S = ^ (gui^+^i+gw/^), etc. Again, Fu = g4 a g v F* = g ii gu F*\ etc, i.e., ^4i = gu(gnF* +gi 2 F 42 +g 13J F 43 ) and two similar equations for F i2 , F 43 . Whence, by (89), gu Mi=- ~ (gnBi+gi 2 B 2 -\-g 13 B 3 ), etc. V-g Et= p^ignDx+guDz+gM, etc, V-g ^Analogously to the sense in which 'the system- velocity' of light was used previously, and contrasted with the local point of view. Light in Gravitation Field 117 or, solving for the polarisation components and noticing that l/g«= A' 44 , Dy= -V^g. g u (g u Ei+g*E a +g»E a ) l etc. Thus B is exactly the same linear vector function of M as is D of E. Introducing the symmetrical linear vector operator (101) £ u gM gM — u> = 2l S a- 2 a 23 gtl „32 ? 33 we can write shortly B = M M, d=a:e, where tl = K = V-g. g «. (102) (103) In absence of gravitation the g M assume their galilean values, the operator d> becomes an idemfactor, g 44 =l, and p=K = l, giving B = M and D = E. From the system-point of view the vacuum is thus transformed by gravitation into a crystalline electromagnetic medium with anisotropic permeability m a "d permittivity K. These operators have, however, by (103). at every point common principal axes (which are orthogonal) and the same principal values. Now, owing to this peculiars ty the velocity of propagation of an electromagnetic wave, although varying from point to point and dependent upon the direction of the wave-normal, can be easily proved to be independent of the orientation of the light vector D. Thus. although the medium is anisotropic, there will be no double refraction due to the gravitational field.* In fact, if n be the wave-normal and t> the velocity, that is the system-velocity *Cf. in this connection A. O. Rankine and L. Silberstein, Propagatioti of light in a gravitational field, Phil. Mag., vol. 39, 1920, p. 586. 118 Relativity and Gravitation of propagation, along the wave-normal, we have from the electromagnetic equations (^4), (B), (102), wilh p = 0, — iOS = VMn, - l xM= FnE, (104) as will be seen at once by considering a wave of discontinuity and using the general compatibility conditions given else- where.* Now, since the operator K is identical with y, the last two equations give - KE+Vn(K- 1 VriE) = 0, for every direction of E. Here the operator K ' 1 is the inverse of K. If Ki, etc., be the principal values of K and «i, etc., the components of n or the direction cosines of the wave-normal with respect to the principal axes, the last equation gives at once &_ nKn = K ini *+K 2 n 2 2 +K 3 n 3 2 i.e., a propagation velocity independent of the orientation of the light vector, which proves the statement. If gi, g 2 , g s are the principal values of the vector operator gn, gn, • • • g33, the inverse of -co, then the principal values of -co itself are 1/gi, etc., and we have, by (103) and since g = glg2g3gU, (105) c 2 - Cwr . n 2 - . n£ \ — + — + — • gl g2 g3 - 1 Such being the formula for the velocity of propagation on the electromagnetic theory of light, it is interesting to com- pare it with the light velocity v yielded directly by Einstein's fundamental equation ds = 0. This velocity is taken along 'the ray' instead of the wave-normal. Thus, by (100), if u be a unit vector along the ray, and u t its direction cosines, *L. Silberstein, Annalen der Physik, vol. 26, 1908, p. 751 and vol. 29, 1909, p. 523, or Theory of Relativity, London, 1914, p. 56. PONDEROMOTTVE FOW E 119 < r/x; dx k — = - gik r- — = -iik*iVk, v da da and especially it //, be the direction cosines with respect to the principal axes of the operator g n , ga, . . . gu, c 2 1 — = ki«i 2 +gs«2 2 -fg»«g*J- (106 ) v 2 gu Formula (85), used in connection with the bending of rays around the sun, is only a special case of (106). In that case the principal axes are along the radial and all the transversal directions, while the principal values are 1 1 gl=— - =— ~ , g2=g3=—l, g*i g* and u l 2 = cos 2 rj, w. 2 2 +W3 2 = sin 2 ?7, so that (106) reduces to (85). If the wave-normal n coincides with a principal axis, say with the first one, we have, by (105), t) 2/ c 2 = — g«/,gi, and by (106), c 2 /v 2 = — gi/gu', that is to say, v = b, as it should be. For then the ray falls into the wave-normal. But in general the ray does not coincide with the wave-normal, and so does v differ from D. The question whether the null-line equation (106) is always compatible with the electromagnetic equation (105) may be left to the care of the reader. If the ray be defined, as usual, by the Poynting flux of energy, its direction will be that of the vector product ITSM, and all questions concerning the light ray will follow from (104) with K = n as given by (103). 39. Ponderomotive force, and energy tensor of the electro- magnetic field. The general tensors corresponding to these were easily suggested by the results already known from the special relativity theory. The inner product of the magneto-electric six-vector and the four-current, i.e., the covariant vector P, = F lK C\ (V) gives the ponderomotive force on a charge, per unit volume, together with its activity or, in other words, the momentum and the energy transferred, per unit volume and unit time, from the electromagnetic field upon the electric charges. 120 Relativity and Gravitation In fact, using for instance cartesian coordinates and g = — 1 , we have for the first three components of P t , by (89) and (91), Pi = P T— (v 2 B 3 -v 3 B 2 ) +E^, etc., or if Pi, P 2 , P s be condensed into the three-vector P, P = P |~E + - FvB~] which is the familiar formula for the ponderomotive force, while the fourth component becomes P 4 = - - (E^+Em+Em) = --?- (Ev) c c or, since FvB = 0, Pi= - — (Pv) c which, apart from the factor — 1/c, is the activity of P. Somewhat more generally, the same formulae will hold with p replaced by p/^ — g . But it will be understood that from the standpoint of general relativity the master formula for the electromag- netic momentum and energy transfer is again (V), as were before the electromagnetic field equations, all generally co variant. By means of (IV) and (V) the four-force P L can be repre- sented as the covariant derivative of a second rank tensor, a generalization of the array of maxwellian stresses, momen- tum and energy density. Following Einstein's example it will be enough to give here the required form of P t for such co- ordinates for which g= — 1, and therefore, by the second of (IV), c k = ii£_- Thus, by (V), p= JL(F lK F K *)-F*^ . ENERGY Tensor 121 The second term is, by the first of the equations (IV), p* dF^ = _ p X S dF A + dF u \ dx x \ dx, dx K ' dx L L dx l dx K J But the bracketed expression vanishes. In fact, since the summation is to be extended over all k, X, and since both /*'- tensors are antisymmetric, this expression can be written p*/dF* | *flu dF u \ V dx, dx K dx x / to be summed only over k < X. But the third term of the bracketed factor is -\-dF lK /dx x , so that the whole factor of F kX vanishes, by the first of (IV). Thus r^xd/^. _ , r,«x ^a _ a Ka x/s E- ^a t ~ » * 2 g g SaP —— , dx x d.r t d.Y t and since here k, X can be replaced by a, /3 and vice versa, p^ep. = -± g "o^JL(F af} F KX ) dx x dx t - "I r^ OF* **) + i *".* ^x f- GT S x ')- dx t dx t The last term can be transformed into — h F Kr F kX g xp dg />r /dx i , so that P, = -L (fl. F« x ) - | -A (F KX F* ) - 1 F* P* ^ ^r . Finally, if we denote by F the invariant F kX F kX and in- troduce the mixed tensor Ti =$F8] - F Ka F*\ (107) the last formula becomes p t =— " -hg KX — n , (ios) dx a dx l 122 Relativity and Gravitation exhibiting the four-force in terms of Tf, the energy-tensor of the electromagnetic field. To recognize in the latter an old friend consider a galilean domain and use cartesian coordinates. Then, the g lK being constant, (108) reduces to the familiar equation dT a dx a and since, by (89), in the present case, F«z = F 23 = M\, etc., Fu= — i^ 4 = Ei, etc., we have F=F af }F a P = 2(M*-E?), and (107) gives 7V = (£i 2 - W) + {Mt - W 2 ) , r 1 2 = r 2 1 =£i£ 2 +-Mi-M'2, etc., which are Maxwell's electromagnetic stress components,* further Ti* = 7V= - (E 2 M 3 -E3M0), etc., which are the negative components of the energy flux divided by c, or the components of electromagnetic momentum per unit volume, and finally which is the negatived density of electromagnetic energy. The right hand member of (108) can be shown to be the divergence of the mixed tensor T* or its contracted covariant derivative T" a as defined by (47a) . In fact, since for constant g, by (67), < V = 0, the said divergence reduces to T '- = ~ J {'/}**' < 42 ^ and since in our case Tp is symmetrical, this can be shown to be identical with Tt a =-± +*?-r Al (42c) dx a dx t where T kX = g KV Ty. On the other hand, since g* x g KV is a constant, to wit 5„ , we have f 9 J- =-gJ-£- dx t dx t [a relation to be used also in passing from (42o) to (42c)], and ^Tensions proper being counted positive. Energy 1 ensor [23 the second term of (10S; becomes identical with th< term of (42c). Thus P t = T^ = Div (77) (10 exhibiting the four-force as the divergence of the mixed energy - tensor of the electromagnetic field. If the electric charges are under the exclusive control of the electromagnetic field, the total four-force P, vanishes, and we have 7? ^Div (77) =0. (109) These four equations are perfectly analogous to the 'equations of matter', (65), given in Chapetr IV, the 'tensor of matter' being now replaced by the energy-tensor of the electromagnetic field defined in (107). These equations express in either case the principles of energy and of momen- tum. Instead of the mixed tensor (107) we can introduce the covariant electromagnetic tensor g lv T* = T lK . If the form (III) of the gravitational field-equations be used, then in the presence of an electromagnetic field the components of the latter tensor (multiplied by the gravitation constant) have to be included in the corresponding components of the tensor of matter appearing in the right-hand member of those equations. Thus both kinds of stresses, energy', etc., con- tribute to the curvature tensor G lK and through k codetermine the gravitational field. The contributions of the electro- magnetic tensor components are, of course, for all technically obtainable fields, exceedingly small as compared with those due to matter in the narrower sense of the word. Theoreti- cally, however, the roles of the two kinds of energy-tensors are equivalent. —9 APPENDIX. A. Manifolds of Constant Curvature. As was mentioned in Chapter III, an w-dimensional manifold of constant isotropic riemannian curvature K, posi- tive, nil or negative, is characterized by the differential equations (54), which can be deduced from the general formula (53) for the riemannian curvature.* If we put where R may be any constant, imaginary or real, finite or infinite, the said equations are (tX, hk) = — (g tll g Xlc - g M g XM ) , (1 10) to be satisfied for all t, X, k, fx. In order to pass from Riemann's covariant symbols to the mixed curvature tensor use (50a). Thus, multiplying both sides of (110) by g Xa and taking account of (32) , a. Einstein's tensor G^ is the contracted curvature tensor G« = - (8 a K g ia -Kg*). JKr The first term in the brackets is simply g x , while the second, *Cf. also W. Killing, Die Nicht-Euklidischen Raumformen, Leipzig 1885, Section 123. 124 Co.VSTAVl CURVATT/WB 121 in which <5* or 1 is to be taken n times, is equal nf^ . Thus, for a manifold of n dimensions, of the said kind, ^ = - n ^- 1 .^, (Ill) for all values of i, k. In fine, the contracted curvature tensor is proportional to the metrical tensor g u . For a three-space the constant factor is — 2/R 2 , and for a four-manifold —3 R 1 , and so on. Notice that we are dealing here with isotropic manifolds, — a remark which will be of importance in the sequel. The curvature invariant is G = g" c G u , and since g 4 * g u =;z, we have, by (111), G= _n(n-1)_ This justifies, in general, the name of 'mean curvature' mentioned in Chapter IV and given to G by some authors. For a three-space we have G=-- , (112 s ) R- and for a four-fold, provided always it were isotropic, we should have 12 G=--. (112<) R 2 It was known for a long time that the line-element of a three-space of constant curvature l/R 2 is, in polar coordinates Xi, x 2 , x 3 = r, 0. 6, d<r*=dr*+R* sin 2 — . [j0 2 +sin 2 <t>dd-]. (113) R In fact, availing himself of (75), the reader will find for (113) , as the only surviving components, 2 r Gn= — — , 6 , 22 = — 2 sin 2 — , G 33 = — sin 2 <t>G 2i , R 2 R that is to say> <?«=—!**' (uis) R- 126 Relativity and Gravitation thus verifying (111) for the case w = 3, whence also G = — Q/R 2 , as above. Manifestly, if we took for da 2 the negative of (113), or 2 inverted the signs of all g ik , we should have Gu= + — ■ g,-,-. Now, it will be well to notice that the same is the case if we subtract the (113) -value of da 2 from the squared differential of a fourth coordinate multiplied by a constant; that is to say, for a four-dimensional manifold defined by ds 2 = dx i 2 -dr 2 -R 2 sm 2 — . [dc^'-fsin 2 ^ 2 ] (114) R we have [not (111) with w = 4 but] 2 G«= — (*=i,2.3); G 44 = 0, as the reader can verify explicitly, and therefore, G = ±G U =+® . In fine, for a four '-fold, say space-time, of the type (114) the three curvature components and the invariant G have the same values as for an isotropic three-space with changed signs. Notice that this result does by no means clash with the general equations (111) and (112). For the space-time determined by the line-element (114) is not isotropic with respect to its riemannian curvature, even if x 4 be replaced by V-l Xi. The latter line-element plays an important r61e in Ein- stein's recently modified theory of which a brief account will be given in Appendix, B. Consider the four-fold defined by the somewhat more general line-element ds 2 = g A dx A 2 -dr 2 -R 2 sin 2 — [d0 2 +sin 2 <^0 2 ], R where g 4 , written for g 44 , is a function of r alone. Then, with /z 4 = log g 4 , the only surviving G-components will be Gn = l (/*7 + 2/i/') - CONSTAM ( I K\ \ II I' I 2 l_'7 !<■■ Goo — I sin 2 <£ C?88 — sin - — 2 Mil- 2r R (115 _1 G 44 = -i (&V+ 2/? 4 ") - — - n>t - gi R R whence the curvature invariant G = —G\i-\-2G 2 2 go + Gu/g*, with g 2 = -R 2 sm 2 (r/R), G= 1 _i(/ ? v+2// 4 ")-— 4 cot - . R* R R Let us now require that G should he constant (which is, at any rate a necessary condition for G lK : g u = const.). Then the last formula will be a differential equation for // 4 = log g+. Now. this equation can he satisfied by g 4 = cos 2 ar, where a is a constant. In fact, this assumption gives h'i = —2a tan ar, h" A = — 2a 2 /cos 2 ar and reduces the last equation to n ., , 4a r „ G 2a~-\- — cot — . tanar = G — — =const., R R R 2 and this equation can only be satisfied either by a = 0, i.e.. g4 = l, and R 2 which leads to the line-element just considered, or by a — 1 R. i.e., gi = cos 2 (r/R), and (7 = 12 R 2 . which gives the line- element ^ 5 2 = cos 2— . dx^-dr-R 2 sin 2 — [dtf+sin-Qdd*], (116) R R utilized by de Sitter. (Cf. Appendix, C, infra). The con- 128 Relativity and Gravitation stant value of the invariant G is in this case R 2 that is to say, apart from the changed sign, such as would correspond, by (112), to a genuine isotropic four-fold con- sidered at the beginning. Moreover, introducing g 4 = cos 2 (r/R) into (115), we have at once 3 1 r 3 G n = - — , G 22 = —— Gss= ~ 3 sin2 v > Gii = vz g4 ' R 2 sm z (f) R R A and since g\ = — 1 , g 2 = — i? 2 sin 2 (r/R) , and all components with i9^k vanish, ^ = | 2 ^, (H6 1 ) which, apart from the changed sign of the constant factor, agrees with (111) for w = 4. On the other hand, substituting into (115) the alternative solution g 4 = l we have, for the line-element (114), 2 G u = — gu («=i,2,3) ; Gu = 0. (114 1 ) The best way of stating the properties of the two solutions is to write the corresponding contravariant tensors which in our case reduce to G a = GJg u . These are, for the line- element (114), QU == Q22 =G S3 = £ G 4 * = 0, (H4 2 ) R 2 and for the line-element (116), G^ = G 22 = G ss = G ii = - . (116 2 ) R 2 Thus the time-space defined by the line-element (116) behaves, apart from the common sign change, as an orderly four-fold of constant and isotropic riemannian curvature. This is its characteristic difference from the manifold defined by (114) which is deprived of isotropy and is a rather loose, uneven melange of time and space. Such at least would be Einstein's \j.\\ Equations L29 the comparison of Einstein's line-element 1 1 1 1 1 with deSitfc (116), from the standpoint of general geometry. Their physical merit must, of course, be judged by other standards. B. Einstein's New Field-Equations and Elliptic Space. About two years after the publication of the original form of the gravitational field-equations, (III), Chapter IY. Einstein found weighty reasons for slightly modifying them.* Without attempting an exhaustive discussion of all his reas for that change or amplification we shall give here a brief account of his new field-equations and of some of their consequences. The tensor of matter T^ being given, the metrical and at the same time the gravitation tensor components g u are not, of course, determined by the field-equations alone, as indeed would be the case with any other set of differential equations in infinite space (and time). A necessary supple- ment of the data consisted, exactly as in the case of Laplace- Poisson's equation, in prescribing the behaviour of the g„ at infinity. Now, as may best be seen from the example of the radially symmetrical field treated in Chapter V ', the g u were assumed to tend 'at infinity', that is, for ever growing r L. to their galihan values g lK , say in cartesian coordinates, -10 0-1 0-1 1 But such boundary or limit conditions, not being independent of the choice of the coordinate system, have seemed ' repugnant to the spirit of the relativity principle'. In fact, to remain generally invariant the limit tensor would have to be an array of sixteen zeros. Moreover, the adoption of the galilean or inertial tensor at infinity would be tantamount to giving up the requirement of the relativity of inertia. For whereas the inertia or mass of a particle generally depends upon the *A. Einstein, Kosmologische Betrachtungen zur allgenieincn Rclativitiits- theoric. Berlin Sitzungsberichtc, 1917, pp. 142-152. 130 Relativity and Gravitation g LK and these are even at the surface of the sun but slightly different from g lK , the mass of the particle at infinity would differ but very little from what it is near the sun or other celestial giants. In fine, the bulk of its mass would be inde- pendent of other bodies, and if the particle existed alone in the whole universe, it would still retain practically all its mass. As a matter of fact we do not know whether such would not be the case.* But somehow, not uninfluenced by Mach's older ideas, Einstein inclines to the belief that every particle owes its whole inertia to all the remaining matter in the universe. Yet another reason against the said conditions at infinity is given which is based on considerations borrowed from the statistical theory of gases and which would equally apply to Newton's theor)^. But for this the reader must be referred to Einstein's original paper (I. c, §1). In conclusion Einstein confesses his inability to build up any satisfactory conditions at infinity, in space that is.f But here a way out naturally suggested itself. The conditions at infinity being hard or perhaps impossible to find, let the world or universe be closed in all its space extensions. If this be a possible assumption, no such conditions were needed. Thus Einstein comes to assume space to be a finite, closed three-fold of constant curvature, in short an elliptic space, either of the antipodal (spherical) or of the polar, properly 'elliptic', kind. But, as we saw before, the curvature proper- ties of space-time are modified by the presence of matter, the invariant G, for instance, being proportional to the density of matter. Thus the curvature of space, as a section of the four-fold, can only be approximately constant and isotropic, and Einstein assumes therefore that space is elliptic or very nearly so on the whole, deviating here and there, within and near condensed matter, from the average value of its curvature 1/R 2 and from isotropy, somewhat as, in two dimensions, a *Provided, of course, we had some massless phantoms to serve us as a reference system and thus to enable us to state the lonely particle's perse- verance in uniform motion. t'Fiir das raumlich Unendliche'. There is nowhere a mention of the behaviour at infinite past or future, no doubt, because such questions with regard to time are not urgent in the usual (stationary) type of problems. Einstein's Ne^j i" r itkm L31 slightly corrugated or wrinkled sphere. As we know, the lino- element of such a three-spare i- da* =dr*+R* kin* — (d^+sinfyd**), and Einstein constructs the line-element which is to determine the four-world 'on the whole' by simply subtracting da'- from dx A 2 = c^df 1 . In short, far enough from condensed matter, stars, planet-. and so on, his line-element, in polar coordinates, is ds- = dx 4 ---dr' 2 -R- sin 2 — (</<£- + si n 2 0</0-). (114) a differential form treated in Appendix A.* Now, this line-element is incompatible with Einstein's older field-equations (III). In fact, the corresponding curva- ture tensor consists of the only surviving components G\,= -|- , G 4 4=0; G= A, (114') *From the four-dimensional point of view, the assumption that three- dimensional space is elliptic is, of course, as unsatisfactory as the older assumption of galilean g lK at infinity. For although the space properties as defined by da 2 are invariant for transformations of the .v 1( xj, .v 3 alone into any x\, x'o, x' 3> they cease to be so when all four coordinates are freely transformed. What is then invariant are the curvature properties of the four-fold of which the three-space is an arbitrary section. * If at least the four-fold (114) were isotropic, Einstein's elliptic space could be invariantly defined as that of its sections to which corresponds the minimum mean curvature, and this is the mean curvature of the four-fold itself (cf. W. Killing, Inc. tit., pp. 79-83). But the four-fold defined by (114) is by no means isotropic, as was explained in A. Figuratively, and with some licence, it resembles not a sphere but rather the surface of a circular cylinder. By (114) not only the value of the curvature of three-space remains unsettled but even its property of being at all a closed space. In fine, the assumption that three-space is elliptic should be as 'repugnant to the spirit of relativity' as was the older condition at infinity. But as a matter of fact it did not appear to Einstein in that light. The clearest way of stating Einstein's new assumption is to say that, outside of condensed matter, it is possible to choose a coordinate system in which the line-element ds 2 assumes the form (114V 132 Relativity axd Gravitation and if these values be introduced into the field-equations (Ilia), which are identical with (III), the result is 1 _ 8tt 3 _ St „ But 'on the whole', that is, outside of condensed matter, T u , T 2 o, T 33 are to vanish (though the value of T i4: and T = p need not be prejudiced), and since actually g n = —1, etc., the in- compatibility of (114) with (III) is manifest. Such being the case. Einstein is driven to modify his original equations (III) by subtracting from their left-hand members the terms Xg„. with a constant X. Thus his new field-equations are G IK -\g Ui =--AT tK -^g lK T) i (117) c- and since these give, obviously, G-4X= — T, (117a) c- they can also be written 5- G x -i(G-2\) gtK = -^ T«. (1176) Since the supplementary term \g lk is itself covariant of rank two, the general covariance of the new equations is manifest. It remains to evaluate the constant X in terms of the curvature 1/R 2 . Now, if we assumed that outside of 'con- densed matter' there is no matter at all, i.e., T^ = for all i, k, we should have, by (114 1 ) and the first of (1176), \ = 1/R 2 , clashing with (117a), through (114 1 ) which would require X = 3/i? 2 . But, as Einstein expressly states, his new theory is to be associated with the approximate concept that all the matter of the universe is spread uniformly over immense spaces. In other words, Einstein substitutes for the granular structure of the universe (the grains being not only planets but stars, nebulae and similar giants) a macroscopically homogeneous distribution of matter, exactly as for many purposes a con- En Xi-.u Eouatio tinuous homogeneous medium is substituted for an assem- blage of molecules or atom-. The total n taiued in universe being M and its volume V, Einstein's homogi • density, prevailing on the whole. Is M Po= — • V Only here and there, within the celestial bodies, the density p exceeds p considerably, and is perhaps somewhat larger in interstellar spaces within our galaxy than half way between one star cluster or 'island universe' and another, a million or more light years apart. Moreover, basing himself upon the known fact of the small relative velocity of stars as compared with the light velocity, Einstein makes the approximate assumption that there is a coordinate system, relatively to which matter is on the whole permanently at rest, and in which therefore the tensor of matter is reduced to its 44-th component which is then also its invariant T = p. In fine, we have outside of condensed matter T ii =T = p as the only surviving component, and therefore, by (414 and (1176).' 1 4- Thus, Einstein's new field-equations 117' become ulti- mately G m -l(c-Q 5lK =- ??r„. (117c) At the same time we see that the curvature of space on the whole is proportional to the average density of matter. 1 4tt ^ = ^p . 118 The whole volume of elliptic space of the polar or properly elliptic kind being 134 Relativity and Gravitation the total mass of the universe, in astronomical units, will be M=—R, (119) 4 which moved some authors to the enthusiastic exclamation: 'the more matter, the more room'. The corresponding 'gravitation radius', or better, the mass in bary -optical units, which is a length, would be L— — = — , (119a) c 2 4 or just one-quarter of the total length of an elliptic straight line.* According even to our coarse knowledge of the average density of matter (some thousand suns per cubic parsec), and in view of the formula (118), it is impossible to believe in a curvature radius much smaller than 10 12 astronomical units or, say, R = 10~° kilometers. This would mean, by (119a), a total mass amounting again, in bary-optical units, to almost 10 20 kilometers. To this tremendous total our own sun contri- butes but 1| kilometers, and our whole galaxy not more than 10 10 kilometers. The total would thus require 10 10 such galaxies or Shapley's 'island universes'. All these stellar systems may perhaps be found among the spirals. But if Shapley's esti- mate (Bull. Nat. Res. Council, 1921, No. 11, The Scale of the Universe) be materially correct, these island universes are from 500 thousand to 10 million light years apart, and then it remains to be seen whether the last mentioned space would be ample enough. Yet it would certainly be foolish to deny the possibility of a much larger R and of the existence of many more island universes. That Einstein's requirement, at least in the present state of astronomical knowledge, can at any rate be satisfied, is perhaps best seen from its form (118) which is compatible with as small an average density as we please. *The total length of a straight line (geodesic) in the polar kind of space is ttR, and in the antipodal or spherical kind of space 2ttR. The total volume of the latter space is 2t 2 R s , which would give the double mass, as in Einstein's paper. The space in question being thus far defined only differentially, the choice between the polar and antipodal kind remains free. Sitter's Space-Time 135 Further details concerning these cosmological speculations will be found in de Sitter's third paper on Einstein's Theory of Gravitation,* where the role played by elliptic 9pace in astronomy since the time of Schwar/.schild (1900) is discussed. The light rays corresponding to Einstein's line-element (114) turn out to be straight lines in elliptic space, and these lines, described with uniform velocity, are also tin orbits of free particles. Planetary motion would undergo some modi- fications due to the finite value of R; but these are, for the present, too small to be detected. Nor does Einstein's 'cosmological term', as the supplement f^/J? 2 to his original field-equations is called, lead to any other predictions verifi- able in our days by experiment or observation. C. Space-Time according to de Sitter. Returning to Einstein's amplified field-equations (117) let us assume, with de Sitter, that there is outside of 'con- densed matter' no matter at all, so that in such domains all the components of T tK , including T i4 , vanish. Thus we shall have, in free space, so to speak, for all l, k. Now, as we saw in Appendix B, these equation- • which are of the form of (111), can be satisfied by the line- element (116), and give G = 12/R 2 . And since, on the other hand, we have _ 3 ~ R 2 ' This is the solution of the cosmological problem proposed by de Sitter in his last quoted paper. Thus, de Sitter's free space-time is defined by the line-element ds 2 = cos-- c-df 2 — dr* - R 2 s'ui 2 --[d4>' 2 + sm 2 4>dd 2 ] (116) R R and is therefore, as we saw, a manifold of constant isotropic *W. de Sitter, Monthly Notkes of R.A.S., November 1917. 136 Relativity and Gravitation curvature. Within matter Einstein's new equations, with X = 3/i? 2 , are valid, i.e., G lK -^-=--(T lK -^g lK T). (120) R z c l The isotropy of de Sitter's space-time, expressed by R 2 as in (116 2 -), distinguishes it characteristically from Einstein's space-time. This goes hand in hand with p = outside of matter proper. De Sitter's line-element differs from Einstein's by g 4 4 = COS 2 — instead of gu = 1- Consequently, if the permanency of atoms be assumed as in Chapter V, the spectrum lines of distant stars should be displaced towards the red. If r be the distance of a star from an observer placed at the origin of coordinates, the observed wave-length should be increased from 1 to 1 : cos — , becoming infinite for r = - — R, the greatest distance R 2 possible in a properly elliptic space. Manifestly, everything is at a standstill at the equatorial belt, i.e., all along the polar of any observing station as pole. This, though sounding strangely, entails no actual difficulty at all. As to the spec- trum shift of less distant celestial objects, de Sitter quotes the helium or 5-stars which show a systematic displacement towards the red such as would correspond to a velocity of 4'5 km. per sec. If, as de Sitter suggests, one-third of this is considered as a gravitational Einstein-effect, the remainder may be accounted for by the decrease of £44, and since the average distance of the I?-stars is believed to be r = 3.10 7 astronomical units, we should have 3 - 1()7 in -6 1— cos =10 5 , R and therefore a curvature radius i? = §10 10 . But there is, for the present, nothing cogent in the attribution of the said Gravitation and El» rao 137 remainder of spectrum shift to the dwindling oi g« witii dm distance, and ii would certainly be premature either to n or to accept the results of this attractive piece of reasoning. Other consequences of the theory and a more thorough comparison with Einstein's solution will be found in de- Sitter's paper. Here it will be enough to mention still that according to de Sitter's line-element the parallax of a star should reach a minimum at t = \tR, the greatest distance in the polar kind of space (which de Sitter prefers to the anti- podal). This minimum, of the semi-parallax, is equal to p = a/R, if a be the distance of the earth from the sun. On the other hand, Einstein's line-element gives, for r = ?,7i\ff, a vanishing parallax. Since de Sitter's minimum is at least as small as £ = 10 -10 = 0"a00002, one cannot reasonably hope to discriminate between the two proposals by direct observations of parallaxes, while indirect ones contain too many assump- tions to be considered as crucial. Soon after the publication of de Sitter's paper Einstein raised some objections to his form of the line-element. For these, however, not altogether crushing, the reader must be referred to Einstein's own paper (Berlin Siizungsberichte, March 1918, pp. 270-272). D. Gravitational Fields and Electrons. The problem of the equilibrium of electricity constituting the electron as the structural element of matter proper, already attacked by G. Mie and others, has been taken up by Einstein in a paper of April 1919 (Berlin Sitzungsber., pp. 349-356). The result of the investigation is that this tempting question cannot be completely answered by means of the field -equations alone. For details of the reasoning the reader must be referrred to the original paper. It will be enough to mention that the fixed relation between the universal constant X in the amplified field-equations and the total mass of the universe, as related in Appendix B, is here given up. Space continues to be considered as closed but the curvature radius R and, therefore, the volume of the universe appears as independent of the total mass contained in it. though its macroscopic density p is still treated as uniform. INDEX (The numbers refer to the pages; Abraham, M Absolute, Cayley's 51,52 Absolute differential calculus. . 39 Absolute value, or size, of vec- tor 51 Angle 50 Antipodal kind of elliptic space 6 Antisymmetrical tensors 44 Associated invariant 53 vectors 54 Astronomical unit of mass .... 74 Atoms, as natural clocks 104 Bary-optical unit of mass 134 Bessel 10 Bianchi, L 28, 39, 64, 71 Boundarv conditions 129 Canal rays Cayley 5 Centrifugal acceleration 3 Christoffel symbols Clifford Closed space Coelostat distortions Compatibility conditions Componens of a tensor Conj ugate vectors Conservation laws Constant curvature Constant light-velocity, prin- ciple Contraction, of mixed tensors. Contravariant devriative tensor, denned . Cosmological term Covariance of natural laws. . . . Covariant differentiation tensor, defined Current, four- Curvature, gaussian 1 riemannian 105 1,52 1,37 59 27 13 130 102 118 41 54 87 67 46 59 41 135 23 59 41 109 ',65 67 Curvature invariant ... Curvature tensor 63,68 contracted 70 Cyanogen spectrum lines 104 Deflection of light rays 100-102 Density of matter 134 Differential geometry 5 quadratic form. . . 14 Differentiation of tensors . . 59 et seq. Divergence of mixed tensor. . . 83 of a vector 62 of a six-vector .... 62 Eclipse expeditions 102 Eddington, A. S 39, 80, 103 Einstein, 1, 10, 11, 12, 19, 22, 24, 28, 38, 39, 56, 58, 61, 70, 77, S2, 86, 88, 104, 107, 113, 129, 130, 137 Electro-magnetic six-vector . . . 109 Electromagnetic equations .... 106 et seq. stress, momentum, and energy 75 Electrons 137 Electrostatic potential 112 Elementary flatness 13 Elevator 11 Elliptic space 6, 130 Energy, principle of 86-87 Energy tensor 75, 122 Eotvos, R 10 Equation of motion, of a free particle 28 Equivalence hypothesis 12 Equivalent differential forms. . 16 Expansion, of a vector 59 of a six- vector .... (52 Fermat's principle 100-101 Field equations, gravitational 70. 77. $9. 132 140 Index Fixed-star system 1 Four-current 109 Four-index symbols 17, 64 Four-potential Ill Four- vector 4, 40 Fundamental quadratic form . . 52 tensor 52 Free particle motion, and geo- desies 8, 20 Galaxies 134 Galilean coefficients 6, 129 Galileo 10 Gaussian coordinates 39 General relativity principle ... 22 Geodesies 7, 26-28 Gradient, of a scalar field 49 Gravitation, and Christoffel symbols 29 Gravitation law, Newton's. ... 73 Gravitation radius 95 Gravitational field equations, 69 et seq., 89, 132 waves 90 Hamiltonian principle 88 Heavy and inert mass 10 Helium or B-stars 136 Hilbert, D 88 Holonomous transformations. . 16 Homaloidal, or flat, manifold . . 67 Hydrogen nucleus 80 Indices, upper and lower 45 Inertia, induced 130 of energy 74 Inertial systems 1 Inner product, of tensors 42 Invariance, of line-element. ... 16 Invariants 42, 46, 47 metrical 53,57, 62, 79, 88, 125 Island universes 134 Isotropic curvature 67, 125 Jacobian 15 Jeffreys, H 99 Jewell, E. L 104 Keplerian laws 96, 98 Killing, W 64, 124, 131 Kottler, F 107 Laplace- Poisson's equation. .69, 73, 74,79 Laue, M. v 75,76 Law of (algebraic) inertia 16 Levi-Civita, T 39, 41,101 Light propagation, and mini- mal lines 8,20 Line-element 5 Linear differential form 113 Lipschitz's theorem 66 Local coordinates 12 Lor, matrix 75 Lorentz, H. A 88 Lorentz transformation 4 Mach, E 130 Magneto-electric six- vector . . . 107 Mass, astronomical unit of. ... 74 Matter : 74, 75, 77 equations 82,85,123 Maxwell's electromagnetic stress 122 equations 106 Mean curvature 79, 125 Mercury's perihelion motion. . 99 Metrical manifold 50 properties of tensors . 53 Mie, G 10,137 Minimal lines 7 Minkowski 4,75 Mixed tensor, defined 45 Momentum, principle of 86-87 Mosengeil, K. v 74 Natural clocks 104 volume 58 Newcomb G9 Newton 10 Newton's equations of motion. ."6 Node motion, of Venus 99 Non-holonomous transforma- tions 14 Norm, of a vector 53 Index MI Outer product, of tensors. 13 Orthogonal coordinates 113 vectors, define I. 56 Parallax 137 Perihelion motion 95 et seq. Permanent field 70 Perturbations, secular 99 Planetary motion 95 6/ seq. Polar kind of elliptic space ... 6 Ponderomotive force, in elec- tromagnetic field 1 H> Potential, electrostatic 112 four- Ill newtoniau 36 retarded 90 vector- 112 Poynting 74, 75 Principe, eclipse expedition . . . 102 Principles of momentum and energy .- 86-87 Product, inner 42,48 outer 43 Propagation of gravitation . ... 90 Proper time 103 Radially symmetrical field . 92 et seq. Radius, gravitation- 95, 134 of world-curvature . . . 80-81 Rank ,of a tensor 40 et seq. Rankine, A.O., and Silberstein, 117 Reduced tensor 50 Retarded potential 90 Reference frameworks . A, et passim Relativity principle, general. . . 22 special 2 Ricci, G 39,41 Riemann 17, 51 , 64 Riemannian manifold 50 Riemann-Christoffel tensor. .62,63 Rotating system 30-35, 101 Rotation, of a covariant vector 61 Russell, H. N 102 Rutherford 81 Scalar, tensor of rank zero .... 42 Scalar product, of tensors 42 Schwarzschild, K 95, 135 ,-r Shapley . . 134 Shift of spectrum lines. L02-106, 138 Sitter, W.de .33,99, 101, 127, 13ft, 138, 137 Six- vector 44 Size, of a vector 51 Sobral, eclipse expedition \0l Space-like vector Special relativity, recalled ... 1-9 Spectrum shift, due to gravita- tion 102-10.'. due to distance 136 Spherical space 6 St. John, C. E 104 Stress-energy tensor 75 Sum of tensors 41 Sun, gravitation radius of.. .95, 134 Supplement of a tensor 55 Sylvester 16 Symmetrical tensors 4". Tangential world 13 Tensors 39 et seq. Tensor character, criterion of.. 48 Tensor of matter 76, 78 Thirring, H 32 Time-like vector 53 Universe, mass and volume of . 134 Vector 40 Vector potential 112 Velocity of light 25, US Venus, motion of nodes 99 Volume 58 Wave, electromagnetic, in gra- vitation field 118 Waves, gravitational 90 Wave surface 26 Water, curvature in 80 Weight and mass 10 Weyl, H 39, SS, 103,113 World curvature 80 vector 4 Wright, J. E 39 Zodiacal matter 99 D.Van Nostrand Company are prepared to supply, either from their complete stock or at short notice, Any Technical or Scientific Book In addition to publishing a very large and varied number of Scientific and Engineering Books, D. Van Nostrand Company have on hand the largest assortment in the United States of such books issued by American and foreign publishers. All inquiries are cheerfully and care- fully answered and complete catalogs sent free on request. 8 Warren Street - - New York Date Due 3£&twrt^ r It/r/tf VHou\ 9 NOV Qi 1001? iW \)WYl£ '^W, ^ , WvH'^ L. B. OAT. NO. 1187 0C173.55.S518 scm _ 3 5002 00386 1643 The theory of general relativity and gra _£ilD_££Stein TITLE . ty -. * QC 173. S^ S518 108170 Theory of gen . relativity & DATE DUE -i^ow^^ra.v4 : -+—— — — — — 7 io A* — "^ Ar> y 1 1 ,' " '^--fc 16^170 PHYSICS LIBRAR)